The 47-year-old American fired a final-round 65 to finish on 21-under for the tournament.
Higgs, who began the round one shot behind overnight leaders Brian Stuard, James Hahn and Cameron Percy, found himself three behind Cink in the closing stages.
The 28-year-old eagled the 16th to put himself in contention but Cink held his nerve to become the oldest player to win a PGA Tour event since Phil Mickelson at Pebble Beach in 2019 aged 48.
Mickelson finished 11 shots behind Cink on Sunday after his six birdies were offset by bogeys at the seventh and ninth, and a double bogey at the eighth.
Doc Redman moved up 40 places to tie for third alongside Chez Reavie and Kevin Streelman after firing a 10-under 62, matching the lowest round of the tournament. Reporting by Arvind Sriram in Bengaluru; Editing by Peter Rutherford