Inclusion Criteria:

* Previously treated, histologically confirmed CD20+ B cell lymphoma which includes mantle cell lymphoma, follicular lymphoma, chronic lymphocytic leukemia/small lymphocytic lymphoma, hairy cell leukemia, marginal zone lymphoma, mucosa-associated lymphoid tissue (MALT) lymphoma, and lymphoplasmacytic lymphoma. Fine needle aspirates are not acceptable
* Ability to understand and willingness to sign an informed consent form
* Ability to adhere to the study visit schedule and other protocol requirements
* Eastern Cooperative Oncology Group (ECOG) performance status score of 0 or 1, or Karnofsky performance status (KPS) performance status of 60% or greater
* Life expectancy \>= 3 months
* Leukocytes \>= 3,000/uL
* Absolute neutrophil count \>= 1,000/uL
* Platelets \>= 50,000/uL
* Total bilirubin: =\< 2.0 x institutional upper limit of normal (ULN)
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional ULN
* Serum creatinine

* =\< 2.0 x ULN, OR
* Calculated creatinine clearance \>= 30 ml/min/1.73 M\^2 by the modified Cockcroft and Gault Formula, OR
* Creatinine clearance \>= 30 mL/min obtained from a 24-hour urine collection
* At least one measurable lesion according to International workshop lymphoma response criteria. There must be measurable lymphadenopathy to follow with serial exam and/or imaging
* All previous cancer therapy, including radiation, hormonal therapy and surgery, must have been discontinued at least 4 weeks prior to treatment in this study
* Submission of original biopsy for review and verification by participating center hematopathologist
* Disease free of prior malignancies for \>= 3 years with exception of currently treated basal cell, squamous cell carcinoma of the skin, or carcinoma "in situ" of the cervix or breast
* All study participants must be registered into the mandatory Revlimid Risk Evaluation and Mitigation Strategies (REMS) program, and be willing and able to comply with the requirements of the REMS program
* Females of reproductive potential must adhere to the scheduled pregnancy testing as required in the Revlimid REMS program
* Able to take aspirin (81 or 325 mg) daily as prophylactic anticoagulation. (Subjects intolerant to acetylsalicylic acid \[ASA\] may use low molecular weight heparin)
* Must be able to swallow study drugs
* Human immunodeficiency virus (HIV) infected patients are eligible provided they meet all the other eligibility criteria of the study in addition to the following:

* No history of acquired immunodeficiency syndrome (AIDS)-defining conditions other than lymphoma or history of CD4+ T-cells below 200/mm\^3 prior to beginning combination antiretroviral therapy (cART)
* After HIV diagnosis and during treatment with cART, patients should have maintained CD4+ T-cells \>= 350/mm\^3 prior to lymphoma diagnosis. Patents who never immune reconstituted to a stable level above 350/mm\^3 are not eligible
* At time of study entry CD4+ T-cells must have recovered from prior lymphoma therapy to \>= 250/mm\^3
* At the time of study entry the HIV viral load must be undetectable by standard laboratory assay
* During prior lymphoma therapy, patients must not have experienced documented infections attributed to the HIV+ status
* No history of non-adherence to cART and willing to adhere to cART while on study
* Antiretroviral drugs with overlapping or similar toxicity profiles as study agents not allowed:

* Efavirenz not allowed due to potential central nervous system (CNS) toxicity
* Stavudine not allowed due to potential neuropathic effects
* Zidovudine not allowed due to myelosuppressive effects
* Patients must be willing to be followed at a minimum of approximately every 3 months by physician expert in HIV disease management

Exclusion Criteria:

* Any serious medical condition, laboratory abnormality, or psychiatric illness that would prevent the subject from signing the informed consent form
* Any condition, including the presence of laboratory abnormalities, which places the subject at unacceptable risk if he/she were to participate in the study or confounds the ability to interpret data from the study
* Use of any other experimental drug or therapy within 28 days of baseline
* Known hypersensitivity to lenalidomide, thalidomide, obinutuzumab, or mannitol
* A history of erythema nodosum if characterized by a desquamating rash while taking thalidomide or similar drugs
* Any prior use of obinutuzumab or CC-486
* Concurrent use of other anti-cancer agents or treatments
* Known active hepatitis B or C. Patients on suppressive therapy with a negative viral load and no evidence of hepatic damage are eligible
* Uncontrolled systemic fungal, bacterial or viral infection (defined as ongoing signs/symptoms related to the infection without improvement despite appropriate antibiotics, antiviral therapy and/or other treatment)
* Active central nervous system (CNS) involvement by lymphoma, including leptomeningeal involvement
* Active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. A history of hemolytic anemia associated with the lymphoma does not exclude a patient from the study
* Pregnant or breast feeding females. (Lactating females must agree not to breast feed while taking azacitidine)
* For CC-486: History of inflammatory bowel disease (e.g., Crohn's disease, ulcerative colitis), celiac disease (i.e., sprue), prior gastrectomy or upper bowel removal, or any other gastrointestinal disorder or defect that would interfere with the absorption, distribution, metabolism or excretion of the study drug (CC-486) and/or predispose the subject to an increased risk of gastrointestinal toxicity
* Abnormal coagulation parameters (prothrombin time \[PT\] \> 15 seconds, partial thromboplastin time \[PTT\] \> 40 seconds, and/or international normalized ratio \[INR\] \> 1.5)
* Significant active cardiac disease within the previous 6 months including:

* New York Heart Association (NYHA) class 4 congestive heart failure (CHF)
* Unstable angina
* Myocardial infarction