Cink, who ended with a 40-yard chip-in for birdie when heavy rain suspended the second round Friday, returned to hole a 25-foot birdie putt on the 11th and then stuffed his tee shot into 5 feet for another birdie on the 12th.
That gave him a five-shot lead, but a bogey from the bunker slowed him down and he wound up with a 2-under 68.
Cink was at 9-under 131, and had a three-shot lead over Woods and Rory Sabbatini, who each shot 66.
Woods, a three-time winner at Firestone South, finished off this rain delay much better this time.
He returned Friday morning to complete a bogey-bogey-bogey finish. Saturday morning, he made three straight birdies and missed another good chance from 5 feet.
'I wanted to make two birdies coming in and get to 5 (under), so that was a bonus,' Woods said. 'At least I'm still right there.'
David Toms was among 24 players who finished the second round Friday afternoon just ahead of a downpour that made the venerable course unplayable, with puddles around the cup and small streams cutting through the fairway.
He had a 66 and wound up in a tie for fourth at 5-under 135 with Rod Pampling (67). British Open champion Todd Hamilton (67) and Davis Love III (68) were another stroke back.
Jim Furyk was standing over his tee shot on the par-5 15th when he backed off and cautioned one of the caddies about his shadow, which was moving behind his ball.
That was a welcome reminder.
'It's been so long since we've seen the sun,' Furyk said.
Sunshine was in the forecast for the remainder of the weekend, and the World Golf Championship was expected to get back on schedule by the end of Saturday.
Cink was still in good position to become only the fourth wire-to-wire winner at Firestone, and the first since Woods won by 11 shots in 2000.
One race that hasn't materialized is for No. 1.
It looks like Woods will survive for at least another week.
The guys who have a chance to end his 262 consecutive weeks at No. 1 -- Vijay Singh and Ernie Els -- played as if they were still hung over from a tough week in contention at Whistling Straits.
PGA champion Singh, who only needed to finish ahead of Woods here to become No. 1 for the first time in his career, twice missed par putts inside 3 feet and finished another 73 on Saturday. He was at 6-over 146.
Els, whose major championship season was filled only with heartache, made two double bogeys on the back to complete his first round Friday morning at 72, shot 77 in the second round and was 9 over par.
Masters champion Phil Mickelson can't be No. 1 this week -- probably not even this year -- although his major season was marvelous -- a first, second, third and a tie for sixth at the PGA Championship.
He shot 75 and was 5 over at a tournament where he normally can score well in his sleep.
'I've been in that position before where you have a mentally grueling week where you're in contention on a hard course, and the next week you almost have a blowout and find it hard to concentrate,' Cink said. 'I'm sure that's what those guys are having right now, because they're a lot better than 5 over and 8 over par.'
Cink was much better than that over two rainy days in northern Ohio.
He stuffed a 7-iron into 5 feet on the spongy ninth green for birdie to finish his first round Friday morning at 7-under 63, matching the lowest round of his PGA Tour career.
He saved par despite a few wayward drives in his second round, and chipped in for birdie on the 10th hole to get to 8 under for the tournament. Facing a 25-foot birdie putt downstream on the 11th hole, officials suspended the second round because of the rain.
'Nothing went wrong. I guess that's what went right,' Cink said of his round.
Toms turned a wrong into a right when he made double bogey to close out his first round. Losing two shots gave him an earlier start for the second round, and he managed to finish his 66 just ahead of another downpour that flooded Firestone South.
'I guess that double put me in a position to be able to finish,' Toms said. 'Other than my last hole this morning, I got on a roll and played solid golf all day. I just did get finished, and it got pretty nasty out there on the last hole.'
Toms wound up four shots out of the lead, but he had one thing going for him.
'He gets to sleep in,' Cink said Friday evening.
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