Using the Giant Metrewave Radio Telescope (GMRT) we performed deep observations to search for radio pulsations in the directions of unidentified Fermi Large Area Telescope (LAT) gamma-ray sources. We report the discovery of an eclipsing black-widow millisecond pulsar, PSR J1544+4937, identified with the un-cataloged gamma-ray source Fermi J1544.2+4941. This 2.16 ms pulsar is in a 2.9 hours compact circular orbit with a very low-mass companion (Mc > 0.017 Msun). At 322 MHz this pulsar is found to be eclipsing for 13% of its orbit, whereas at 607 MHz the pulsar is detected throughout the low-frequency eclipse phase. Variations in the eclipse ingress phase are observed, indicating a clumpy and variable eclipsing medium. Moreover, additional short-duration absorption events are observed around the eclipse boundaries. Using the radio timing ephemeris we were able to detect gamma-ray pulsations from this pulsar, confirming it as the source powering the gamma-ray emission.