This phase II trial tests the effect of venetoclax in combination with standard of care (SOC) zanubrutinib in treating patients with chronic lymphocytic leukemia (CLL) or small lymphocytic lymphoma (SLL). Venetoclax is in a class of medications called B-cell lymphoma-2 (BCL-2) inhibitors. It may stop the growth of cancer cells by blocking Bcl-2, a protein needed for cancer cell survival. Zanubrutinib blocks a protein called Bruton tyrosine kinase (BTK), which may help keep cancer cells from growing. It is a type of tyrosine kinase inhibitor. Giving venetoclax in combination with SOC zanubrutinib may be safe and tolerable and may reduce the number of cancer cells that remain in the body in patients with CLL or SLL that have not previously received treatment or at least two years have passed since completing initial treatment.