The American looked set to be joined at the top by overnight co-leader Kevin Na, but a double bogey at the last saw Na drop to eight-under.
Kevin Kisner, Bill Haas and Ben Martin are one shot back on nine-under, with a group of six players that includes Na and Sergio Garcia a further shot back.
There was further woe for Tiger Woods, while Rory McIlroy's round of 70 left him four shots off the pace.
Kirk's second consecutive 68 on the TPC Sawgrass course was built on the back of birdies at the fourth, seventh and ninth as the American took the turn in 33. However, bogeys at the 13th and 14th threatened to derail his round. It would prove a momentary blip though, as Kirk rebounded to record three consecutive birdies from the 15th to the 17th.
Na looked on course to match Kirk, but fluffed his lines at the last moment. After beginning his round with a bogey, Na registered birdies at the sixth, ninth and 11th holes before dropping a shot at the 15th. He would grab it right back with a birdie at the 16th, but it all went horribly wrong on the last as Na proved unable to recover from finding the sand and slumped to a double bogey.
Garcia recovered from a frustrating day on Friday to post a 67 which saw him tied for the lead with Justin Thomas early on. The Spaniard snapped his putter in anger following a poor second round, and while his putting still left something to be desired, an eagle on the par-five second coupled with five birdies and just two bogeys saw him climb up the leaderboard.
One man who was very much going in the opposite direction is former world number one Tiger Woods who carded a 75 to drop back to even par and out of contention.
For the first time in his career, Woods slumped to multiple double bogeys at par fives, dropping two shots at both the second and the ninth.
A horrible drive at the second which hit a tree and only travelled 100 yards got the round off to a bad start, with the American requiring five shots just to make it onto the green.
A bogey at the sixth was cancelled out by a birdie at the seventh, but it all went pear-shaped once more at the ninth. This time it was Woods' short game that let him down, fluffing two attempts at flop-shots before using a wedge to get onto the green from where he two-putted.
An uneventful back nine included just one birdie, but saw Woods lose no further ground.
Current world number one Rory McIlroy endured a poor day, but not quite on the same scale as Woods, with the Ulsterman's 70 leaving him on six-under and four shots off the pace.
It was a round of two halves for McIlroy, with birdie at the par-five second signaling a strong start, but bogeys at the fifth and eighth checked his progress. However, McIlroy bounced back to produce a bogey-free back nine that included birdies at the 11th, 13th and 16th.