Cook closed with a 5-under-67 at Hualalai Golf Club to catch Frost at 17-under 199. Cook made a 16-foot birdie putt on the 18th green.
Frost shot a 69, missing a 15-foot birdie putt at the 18th that would have given him a win in the Champions Tour season-opener. He had held the lead alone since the 12th hole of the second round.
"He is just really good and he's played really well, especially the last few months of last year," Cook said of Frost. "He's won a lot of tournaments all over the world. He hit some beautiful golf shots today and didn't get much out of what he was doing and I hadn't either. It was kind of a survival day until it got down to the nitty gritty and you knew what you had to do."
Both birdied the first playoff hole (No. 18) from inside 10 feet and went to the 17th tee. Cook hit his tee shot about 15 feet left of the hole and drained it after Frost had come up short with his 20-foot birdie try.
Cook won $309,000 for his ninth victory on the senior tour, to go along with 11 on the regular PGA Tour. He birdied all four par-5 holes on Sunday, while Frost birdied just one.
"Overall I was in there all day," Frost said. "I hit a lot of great shots, but didn't capitalize on some of the great shots I hit. He (Cook) played really well, didn't miss any make-able putts."
Cook's victory gets him invited back to this tournament of champions next year. He did not have bogey all week while Frost went bogey-free over his last 48 holes.
Bernhard Langer chipped in for a birdie on the last hole to finish third alone at 200. He started six back and bogeyed two of the first three before getting birdies on nine of the next 11 holes and finishing with a 64.
Fred Couples was five back going into the final round and also had two early bogeys. He rallied with an eagle and six birdies. That run died with a three-putt bogey at the 17th and he was fourth after a shooting a 67 and finishing at 202.
Kenny Perry was the first to make a move Sunday. He started the day in 14th place but surged into fifth with six birdies and an eagle. His 64 matched Langer's score as low round of the week. Perry finished four back with Kirk Triplett.