Provided herein are methods of treating cancer in a cancer patient by overcoming resistance of a cancer cell to a drug, or methods of overcoming resistance of a cancer cell to a drug, comprising administering to the patient an effective amount of (i) an EGFR tyrosine kinase inhibitor or a B-Raf kinase inhibitor, and (ii) an HDAC inhibitor. Also provided are methods for inhibiting or preventing proliferation of drug tolerant persister (DTP) cells comprising contacting these cells with an EGFR tyrosine kinase inhibitor or B-Raf kinase inhibitor in combination with an HDAC inhibitor. Yet also provided are methods for inhibiting or preventing formation of colonies of drug tolerant expanded persister (DTEP) cells comprising contacting these cells with an EGFR tyrosine kinase inhibitor or B-Raf kinase inhibitor in combination with an HDAC inhibitor.