Pembrolizumab With or Without Anetumab Ravtansine in Treating Patients With Mesothelin-Positive Pleural Mesothelioma

Inclusion Criteria: * PRE-REGISTRATION * Patients must have histologically or cytologically confirmed malignant pleural mesothelioma * Patient is willing to submit a tissue sample to test for expression of mesothelin * Note: Tissue sample for mesothelin assay may have been collected prior to, during or after receipt of the frontline chemotherapy; patients will not be required to submit another tissue sample after receipt of the chemotherapy * Patients must have received platinum based chemotherapy * REGISTRATION * For phase 2 only: * Patient has measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 for non-pleural disease or modified RECIST 1.1 (mRECIST) for pleural disease * Note: For pleural disease, this is defined as at least one lesion that can be accurately measured perpendicular to the chest wall or mediastinum that is \>= 10 mm (\>= 1 cm); for extra pleural disease, measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 10 mm (\>= 1 cm) for non-nodal lesions and \>= 15 mm (\>= 1.5 cm) for nodal lesions with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam as per RECIST 1.1 * Tissue submitted for testing at pre-registration shows moderate or stronger mesothelin expression in \>= 30% of the tumor cells * For phase 1 and 2: * Patients must have received platinum-based therapy with or without bevacizumab * Eastern Cooperative Oncology Group (ECOG) performance status \< 2 (Karnofsky \>= 70%) * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin within normal institutional limits * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal (ULN) * Creatinine within normal institutional limits OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * International normalized ratio (INR) or prothrombin time (PT) =\< 1.5 x ULN AND partial thromboplastin time (PTT) or activated PTT (aPTT) =\< 1.5 x ULN, unless patient is on stable dose of anti-coagulation therapy in which case patients will be allowed to participate if they have no signs of bleeding or clotting and the INR/PT and PTT/aPTT results are compatible with an acceptable risk-benefit ratio as per the investigator's discretion * Negative serum pregnancy test for females of child bearing potential * Note: Females are considered to not be of child bearing potential if any of the following apply: * Postmenopausal (defined as at least 12 months with no menses without an alternative medical cause; in women \< 45 years of age, a high follicle stimulating hormone \[FSH\] level in the postmenopausal range may be used to confirm a postmenopausal state in women not using hormonal contraception or hormonal replacement therapy; in the absence of 12 months of amenorrhea, a single FSH measurement is insufficient); * Have had a hysterectomy and/or bilateral oophorectomy, bilateral salpingectomy or bilateral tubal ligation/occlusion, at least 6 weeks prior to screening; * Has a congenital or acquired condition that prevents childbearing * Patient agrees to use one of the following acceptable methods of contraception prior to study entry, during study participation, and for at least six months after receiving the last dose of study treatment: * Acceptable methods of contraception are: * Single method (1 of the following is acceptable): * Abstinence, if consistently employed as the patient's preferred and usual lifestyle and if considered acceptable by local regulatory agencies and Institutional Review Boards (IRBs) * Intrauterine device (IUD) * Vasectomy of a female patient's male partner * Contraceptive rod implanted into the skin * Combination method (requires use of 2 of the following): * Diaphragm with spermicide (cannot be used in conjunction with cervical cap/spermicide) * Cervical cap with spermicide (nulliparous women only) * Contraceptive sponge (nulliparous women only) * Male condom or female condom (cannot be used together) * Hormonal contraceptive: oral contraceptive pill (estrogen/progestin pill or progestin-only pill), contraceptive skin patch, vaginal contraceptive ring, or subcutaneous contraceptive injection * Ability to understand and the willingness to sign a written informed consent document, unless patient is of impaired decision making capacity in which case patient may be eligible if they have a legal authorized representative or caretaker available Exclusion Criteria: * Patients who have received any monoclonal antibody therapy within 4 weeks prior to entering the study * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1) * Note: Patients with =\< grade 2 neuropathy or =\< grade 2 alopecia are an exception to this criterion and may qualify for the study * Note: If patients received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy * Patients who are receiving any other investigational agents * Patients with known brain metastases with progressive neurologic dysfunction, requirement of steroids and lack of improvement on head imaging obtained prior to consent to this clinical trial should be excluded because of their poor prognosis and because they would confound the evaluation of neurologic and other adverse events * Note: Patients with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging using the identical imaging modality for each assessment, either magnetic resonance imaging \[MRI\] or computed tomography \[CT\] scan, for at least 4 weeks prior to the first dose of trial treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to trial treatment * Note: Patients with carcinomatosis meningitis should also be excluded * History of allergic reactions attributed to compounds of similar chemical or biologic composition to anetumab ravtansine or pembrolizumab * Patients receiving any medications or substances that are strong inhibitors or inducers of CYP3A4, including herbal preparation containing CYP3A4 inducers (e.g., St. John's Wort), grapefruit and grapefruit juice (CYP3A4 inhibitor), within 2 weeks before the start of study treatment * Patients are prohibited from receiving the following therapies during the screening and treatment phases (including retreatment for post-complete response relapse) of this trial: * Antineoplastic systemic chemotherapy or biological therapy * Immunotherapy not specified in this protocol * Chemotherapy not specified in this protocol * Investigational agents other than anetumab ravtansine and pembrolizumab * Radiation therapy (Note: Radiation therapy to a symptomatic solitary lesion or to the brain may be considered on an exceptional case by case basis after consultation with Cancer Therapy Evaluation Program (CTEP); the patient must have clear measurable disease outside the radiated field; administration of palliative radiation therapy will be considered clinical progression for the purposes of determining progression free survival \[PFS\]) * Live vaccines within 30 days prior to the first dose of trial treatment and while participating in the trial; examples of live vaccines include, but are not limited to, the following: measles, mumps, rubella, chicken pox, yellow fever, rabies, Bacillus Chalmette-Guerin (BCG), and typhoid (oral) vaccine; seasonal influenza vaccines for injection are generally killed virus vaccines and are allowed; however, intranasal influenza vaccines (e.g., Flu-Mist) are live attenuated vaccines, and are not allowed * Systemic glucocorticoids for any purpose other than to modulate symptoms from an event of suspected immunologic etiology; the use of physiologic doses of corticosteroids may be approved after consultation with the study principal investigator (PI) and CTEP * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Women who are pregnant or breastfeeding. * Note: Pregnant women are excluded from this study because anetumab ravtansine and pembrolizumab are agents with the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with anetumab ravtansine and pembrolizumab, breastfeeding should be discontinued if the mother is treated with anetumab ravtansine or pembrolizumab * Human immunodeficiency virus (HIV)-positive patients who do not meet all of the following and/or are on HIV medications considered to be strong inhibitors or inducers of CYP3A4: * Undetectable HIV viral load by standard clinical assay within 6 months of registration * Willing to adhere to antiretroviral therapy that has minimal overlapping toxicity or pharmacokinetic interactions with protocol therapy * No acquired immunodeficiency syndrome (AIDS)-defining events other within the past 12 months * Near normal life expectancy if not for the presence of the cancer * Has a known history of hepatitis B (defined as hepatitis B surface antigen \[HBsAg\] reactive) or known active hepatitis C virus (defined as HCV ribonucleic acid \[RNA\] \[qualitative\] is detected) infection * Note: No testing for hepatitis B and hepatitis C is required unless mandated by local health authority * Patients who are known to have a history of or a finding of corneal epitheliopathy at pre-study are excluded because anetumab ravtansine may worsen this condition and reduce vision * Known additional malignancy that is progressing or requires active treatment; exceptions include basal cell carcinoma of the skin, squamous cell carcinoma of the skin that has undergone potentially curative therapy, or in situ cervical cancer * Receipt of transfusion of blood products (including platelets or red blood cells) or administration of colony stimulating factors (including granulocyte colony-stimulating factor \[G-CSF\], granulocyte macrophage colony-stimulating factor \[GM-CSF\], or recombinant erythropoietin) within 4 weeks prior to study treatment * Patient with active interstitial lung disease (ILD)/pneumonitis or a prior history of ILD/pneumonitis requiring treatment with steroids * Patient has received prior treatment with PD-1, PD-L1 or PD-L2 inhibitor