With four straight birdies and an unlikely 25-footer to close out the front nine, Woods was 3 under par for his round, two back of the leaders and headed for a few easier holes on the back nine with a chance to put himself squarely in the mix for the final round of the Quicken Loans National.
The Italian (who has twice gone up against Woods in Ryder Cup singles) cruised to his first PGA Tour victory and eighth worldwide with a sizzling final round of 62 in searing conditions at TPC Potomac.
Molinari matched the 18-hole course record with the lowest final round by a US PGA victor this season, his career low on the American tour by two shots. Sung Kang of South Korea fashioned a 64 earlier in the day and finished alone in third at 12 under.
Missed birdie opportunities on the first two holes of Woods' back nine were followed by a blemish at the 13th, with the tournament host holing a six-foot gain at the 16th but then failing to get up-and-down from a bunker to save par at the last.
A group after Ancer closed with a nine-foot birdie putt, 14-time major victor Woods found a bunker at 18 and missed an eight-foot par putt, taking his fourth bogey to shoot 68 and share 10th on 203.
Lahiri, who had a fine three-under 67 on first day, needed pars on his last three holes to survive and make the cut.
In 2017, Kyle Stanley climbed the leaderboard to get into a playoff with Charles Howell III and emerged the victor on the first extra hole to earn his second career PGA TOUR win. Woods tied for fourth, his best result since a runner-up finish at the Valspar Championship three months ago, though he was 10 shots behind. The Swede started the day seven shots behind compatriot Marcus Kinhult but he birdied two of his last three holes to finish at 7-under 277, just ahead of Scotland's Russell Knox, England's Chris Wood, and Julian Suri of the United States.
Gareth Southgate makes tournament strategy revelation ahead of England vs Colombia
The route we've got now is Colombia , who have been poor and had James Rodriguez limped out of the Senegal game. I've always said we've had a chance but at the moment I can't see past Brazil.
The roars that echoed throughout the grounds after his chip-in for birdie at 18 on Friday and his run of four consecutive birdies on Saturday were roars that only Woods could produce.
"I guess one word (today) would be frustrating, to make four bogeys, I couldn't afford to do that".
Zac Blair (66) and Ryan Armour (68) shared third place at 11 under.
"I was hitting the ball well", said the 27-year-old.
Woods wants officials to refrain from watering the course heavily overnight, believing he has a better chance of winning if conditions are dry, firm and hard. "You just have to hit the golf ball well there". A birdie on 14 augured well, before he bogeyed 15th and 16th.
"I'm going to have to shoot a low round to try to give myself a chance", said Woods.
"I grew up watching him play when I was in my teens and starting to think about turning pro", Molinari, 35, said of Woods.