Cook and second-round leader Frost each ended regulation play knotted at 17-under-par 199 after Cook birdied the final hole to force a playoff.
Each golfer recorded a a birdie on the first playoff hole, then proceeded to the par-three 17th at Hualalai Golf Course for the second extra link.
Cook set up his victory with a drive that left him with a straightaway lie 10 feet from the pin, while Frost was left with an uphill task.
Frost left his putt short, but Cook was true with his birdie try to wrap up his ninth Champions Tour victory and second in three years at the Hawaii event.
Bernhard Langer claimed sole possession of third place at minus-16 with an 8-under 64 after birdying 10 of his final 15 holes.
Fred Couples (67) finished fourth at 14-under.