Kirk followed up a sizzling second-round 62 with a solid effort on the par-70 PGA National course in Palm Beach Gardens, Florida, putting himself in a good position to strike for a fifth title, but his first since Colonial in 2015.
The 37-year-old American, who took a break from golf in 2019 to deal with depression and alcohol misuse, had a two-stroke lead over 32-year-old Eric Cole, who posted his second straight 66.
Overnight leader Justin Suh carded an even-par 70 and was three shots off the lead on 200 - one stroke in front of Ireland's Shane Lowry and England's Ben Taylor.
"I made a couple really good driver swings kind of towards the end of my round today, especially on 18," said Kirk, who started the day one off the lead and got things going with a 16-foot birdie putt at the fourth.
"I feel like that's going to be an important thing for me tomorrow, to go out and drive it well."
After a six-foot birdie at the eighth and a 27-footer at the 11th he was holding a three-shot lead at 12-under.
He picked up another stroke at 12 before a bogey at 13, and a three-putt bogey from the fringe at the 16th saw his lead dwindle before back-to-back birdies to cap his round.
At the par-three 17th, Kirk rolled in an 18-foot birdie and at the par-five 18th he missed the green, got relief from a cart path and chipped to six feet, calmly sinking the putt."
Cole, a tour rookie who has 50 wins as a professional on various mini-tours, had five birdies in his four-under effort and was looking forward to challenging for a tour title.
"It's a bigger stage and stuff, but the golf ball doesn't know the difference, and I'm just going to try and keep kind of executing the way I have the first three days, and I think it should take care of itself," Cole said.
Suh eagle
Suh's round was highlighted by an eagle at the par-four 12th, where he holed his 153-yard second shot from the fairway.
He had bogeys from greenside bunkers at the fourth and 13th, and missed a birdie putt from inside four feet at the 18th.
"The hole looked pretty small today - couldn't make any putts," he said. "I was hitting it with good speed and just kept on reading it wrong."
Lowry, who led late last year but came up one shot short of winner Sepp Straka, kept himself in contention with a five-under-par 65 that featured seven birdies.
That included a chip-in at the 11th from the collection area below the green that took one hop and rolled into the cup, the flagstick keeping it from racing by.
"I was just trying to leave myself something inside 10 feet for par," Lowry said.
