Woods, whose 79 career US PGA victories are three shy of Sam Snead's all-time record, has not won a title since the 2013 WGC Bridgestone Invitational - a drought of 1,791 days in which 114 different rivals have taken victories.
Zac Blair and Ryan Armour are two shots back of the leaders at 11-under par. And then he opened the back nine with a pair of birdie chances just inside 10 feet and missed the both.
Woods, playing his 11th US PGA event in a comeback from spinal fusion surgery, made four birdies in a row and sank a 25-foot birdie putt at the ninth. And with a birdie on the final hole for a 67, he tied for sixth to earn his first trip to the British Open.
Molinari has five European Tour victories, has played on two Ryder Cup teams and is No.17 in the world rankings.
But Woods was just two shots out of second, a crucial difference as it relates to some of his other goals this year. "It was a lot easier than I thought (it would be)".
If you would have told me one week ago that Tiger Woods would finish in the top 10 in putting out of the 2018 Quicken Loans National field, I would have asked you which seat Woods would be placing the trophy on his private flight back to Florida. Molinari never buckled on the front nine despite hearing the Tiger roars and then took a stranglehold on the tournament with an eagle from 48 feet on the 10th, a birdie from 2 feet on the 11th, a birdie from 7 feet on the 12th, a birdie from 12 feet on the 13th and another birdie from 2 feet on the 14th.
"The putts I missed, I hit a lot of good ones, which I don't mind because I hadn't been doing that for a while, so this was nice".
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Molinari came into the final round in a tie for the lead with Abraham Ancer.
The real story was Molinari, however, whose best finish this year so far had been a tie for 16th place at the Wells Fargo Championship.
Woods posted successive birdies from over 20 feet over his front nine to get within four of the lead, with a tied-fourth finish being his second-best on the PGA Tour this season.
"Basically, just trying to get efficient hitting the golf ball and then getting comfortable hitting the ball down", Woods said.
Despite the setback at the 18th, Woods sounded generally happy with his form on the greens in his first tournament using a new "mallet" putter, so called because of its resemblance to the tool. "Evidently, I would have to shoot 24 on the back nine".
"I'm starting to hit some putts". "That's something I've been missing for a while, really the past two months, so I'm encouraged".
His strong start gave fans hope of a Sunday charge, but ultimately Woods recognizes that nothing he could have done would have denied Molinari.
The Quicken Loans National started out with a bang as 47 players went under par in Thursday's first round. Starting the week at 82nd in the world, Woods is projected to move up to 67th, the highest he has been since early 2015.