Singh played his first five holes in 5 under par, then finished with three straight birdies for a 10-under 61 to set the course record at the TPC of Boston and build a three-shot lead over Woods, who shot 67.
Woods now has won four straight times on the PGA Tour, and Singh is poised to end that streak.
'It would be good,' Singh said. 'I'm not going to be thinking about his streak or beating him. I've been playing long enough to know that you don't go out there and worry about the guys who are playing with you. You worry about your own game and see what happens.'
There were no worries Saturday.
Even more impressive than his score were the miserable conditions in which Singh shot his 61.
A light rain at the start of the third round never let up, coating the fairways with a thin layer of moisture, making it difficult to keep clubs dry. With little wind, however, Singh took dead aim at the flags and was never too far off.
'This is one of the worst conditions you want to play golf in because it gets everything wet,' Singh said. 'I just kept on plodding.'
And he kept making birdies, finishing at 11-under 202.
Woods had a chance to get closer, if not catch him over the final two hours after Singh was done. Woods missed four birdie putts inside 15 feet on the back nine, including putts of 10 feet on No. 15 and 8 feet on No. 17.
He looked out of sync over the closing holes, letting his hand fly from the club even on shots that turned out well. There was frustration even when he reached the par-5 18th in two, setting up a two-putt birdie that put him in the final group with Singh.
Justin Rose also birdied the 18th for a 69, leaving him tied with Woods at 205.
J.J. Henry, three weeks away from playing in his first Ryder Cup, shot 68 and was at 207, along with Shaun Micheel (68).
'Hopefully, I can play a little better than last time,' Woods said of his duel with Singh. 'I just know that tomorrow, given that conditions are as soft as they are, I'm going to have to make some birdies.'
Circumstances were far different two years ago.
Singh was at a level few others have reached, winning nine times in 2004 and setting a PGA Tour record by winning nearly $11 million. Perhaps the most significant victory came at Deutsche Bank, where he beat Woods in the final round to become No. 1 in the world.
Now, the roles are reversed again.
Woods has won six times this year, including his last four starts, and is so far ahead in the world ranking that he probably could take a year off and still be No. 1.
For Singh, it has been nothing but frustration.
He missed the cut in the final two majors and has not finished in the top 10 since the Western Open. His only victory this year was at the Barclays Classic, and Singh still isn't sure how that happened.
But he kept pounding away on the range, and finally reaped the reward in a big way.
'I kept telling myself, and everybody that is associated with me said, 'You're playing well. Just go out there and enjoy it, and it's going to happen.' And it was really difficult to keep practicing and not seeing results,' Singh said. 'And finally it showed up, and I did it in a great way. It didn't feel like it was difficult to do the way I played.'
Singh has been around long enough to know that some low scores can be the result of a few more putts going in.
Not this one. This was pure from the start.
He teed off 90 minutes before the leaders and was atop the leaderboard before Woods stuck a tee in the ground. He hit wedge into 5 feet on the opening hole, hammered a 3-wood over the water hazard to 30 feet for eagle on No. 2, then fired at the flags on the fourth and fifth holes for two more birdies.
Two more bursts followed -- one that gave him the lead, the other that should make him hard to catch on Monday.
Singh was one shot ahead of Woods when he hit 3-iron into 12 feet on the 16th, spun a wedge back to tap-in range on the 17th, then finished his record round with an 18-foot birdie. His previous best on tour was a 62 in the final round at Hartford in 1998.
The 43-year-old Fijian didn't see this one coming.
'Each time I made a birdie, I got even calmer for the next hole,' he said. 'And I just had a great time out there. I hit a lot of great shots and just tried to play hole-by-hole and never look ahead.'
It was difficult not to look ahead to Labor Day and his showdown with Woods.
Woods was still rebuilding his swing with Hank Haney in 2004, about two months away from the pieces falling together. Singh was about to embark on three straight victories, and the hole looked huge.
'It seems so far away, two years ago, the way I was playing then and now,' Singh said. 'I think this golf course ... it brings back great memories, and hopefully, it's the start of another great run.'