There were 38 rounds in the 60s in a tournament that features many of the top players in the world, and is the final stop on the PGA Tour prior to next week's Players Championship.
Those one shot off the lead at 66 included Rickie Fowler, John Senden, D.A. Points, Brian Davis and Patrick Reed. Fowler turned in his lowest first-round score of the year in a bid for his first tour title.
Cink won the British Open three years ago when he defeated Tom Watson in a playoff and deprived Watson the honor of becoming by far the oldest major champion in golf history.
That has been Cink's last victory to date and he is trying to become the second former British Open champ in three weeks to win on the tour. Ben Curtis, the 2003 winner, captured the Texas Open in San Antonio and has continued to play well -- firing a 69 Thursday.
Woods made three birdies and four bogeys in his first appearance since finishing in a tie for 40th at the Masters.
Mickelson, whose two triple bogeys during the week at Augusta cost him what would have been his fourth Masters victory, had a triple bogey Thursday at the par-4 fourth.