Cink posted a 6-under 64 on Saturday to take the lead at 9-under-par 201.
'I was a little surprised because it's Tiger Woods and the way he's playing lately,' said Cink. 'But the way the course is, anybody, even the great Tiger Woods, can make some bogeys out there.'
Woods, the overnight leader, despite hitting over the roof of the clubhouse on his final hole Friday, chipped in for birdie at the third to reach 10 under par and move two ahead of Love.
The wheels came off on the par-3 fifth. His 7-iron tee ball landed 45 feet from the hole, then his birdie try flew 8 feet past the target. Woods missed that putt to fall back to 9 under par.
At the sixth, Woods drove into the right rough, then found a bunker with his second. He blasted out to 6 feet, but missed that putt to fall into a tie for the lead with Love, who opened with seven straight pars.
Woods missed a five-footer for par at seven, then a seven-footer at the eighth. When Love drained a seven-footer for birdie at eight, Woods suddenly trailed by three.
Love continued his strong play around the turn. He kicked in a 4-foot birdie putt at the ninth, and rolled in a 10-footer for birdie at the 10th to get to 11 under par.
Love dropped strokes to par at 13 and 14 and that's when Cink's name appeared near the top of the leaderboard.
Cink, who was named to the U.S. Ryder Cup team by Tom Lehman on Monday, saved an unlikely par at the third, then knocked a 6-iron to 20 feet to set up birdie at the fourth. He made it two in a row with a 10-footer at the fifth and added a 10-footer for birdie at eight to get to 6 under par for the championship.
On the second nine, Cink hit a 6-iron to 20 feet at the 12th and converted the birdie putt. One hole later, Cink hit a pitching-wedge to 8 feet and sank that putt to get to minus-8.
He was still two down to Love, but a chip-in birdie at the 17th and a solid par at the closing hole got Cink into the house at 9 under par.
When Love bogeyed 16 and Woods' two back-nine birdies were not enough, Cink found himself in a familiar position at Firestone Country Club and in a familiar scenario.
Cink went wire-to-wire en route to victory here in 2004. That was also the same week then captain Hal Sutton named Cink as one of his two Ryder Cup picks.
'Any time you're in this situation it feels a little different,' said Cink, who has not won since this tournament two years ago. 'Last time, it was by five shots. I've got a lot more of a dog fight on my hands.'
Perhaps the biggest fight will come from Woods.
After his disastrous bogey run, Woods holed an 8-foot birdie putt at the 13th. He chipped in for birdie at the 17th, then hit another erratic iron at 18, but got up and down for par.
Jim Furyk was among the leaders, but two bogeys in his last three holes cost him. He finished with a 1-under 69 and is alone in fifth place at 7-under-par 203.
Lucas Glover shot a 1-under 69 on Saturday and is sixth at minus-6, one shot better than Ernie Els (70), Kevin Stadler (70) and Adam Scott (71).