But a double bogey on 11 dashed Woods' hopes of capturing his first major victory since 2008.
Playing alongside a rejuvenated Tiger Woods, Molinari followed 13 straight pars with a birdie on the 14th and finished a brilliant round in fitting style with another from just three feet on the 18th.
Both now sit at five under par heading into the final Sunday, with a four-shot gap between them and the leaders Jordan Speith, Kevin Kisner and Xander Schauffele. "I didn't particularly enjoy that feeling", he said.
"I think it's going to be really important to adapt quickly to the different conditions", said Molinari after nearly matching the 64 posted by Briton Justin Rose earlier in the day.
Despite a modest Open record, he had been backed on the strength of a fine few months that saw him claim the PGA Championship at Wentworth - his biggest win on the European Tour - and his maiden PGA Tour victory at the Quicken Loans Classic.
Francesco Molinari walks on to the 16th green during the final round of The Open Championship golf tournament at Carnoustie Golf Links. But I felt really good this morning.
For someone whose self-belief has always seemed unshakeable, the 42-year-old genuinely seemed surprised when it dawned on him that he could have potentially been crowned champion on Sunday after his third-round 66.More news: Fight over parking spot leads to deadly Florida shooting near Clearwater
More news: Sen. Jeff Flake Delivers Rebuke Of Trump-Putin Summit
More news: Epic first Aquaman trailer makes a splash at Comic-Con 2018
Spieth failed to make even one birdie in the final round, adding instead three bogeys and a double bogey, the latter courtesy of that visit to the bush on the par-5 sixth.
McIlroy and Rose were in good spirits after finishing tied second, two shots behind Molinari.
"It's fantastic to stand here with the Claret Jug", Molinari told reporters after a closing 69 secured an eight-under tally of 276 and a two-stroke victory over Rory McIlroy, Justin Rose, Kevin Kisner and Xander Schauffele. It would surely have been a thought that crossed his mind during the opening two days here, when he was so far down the leaderboard that he would have needed binoculars to see the top.
Justin Rose, who made the cut with an 18-foot birdie putt on the final hole of the second round, is 5 under through 15 holes and 2 under for the tournament. Just keep it in perspective, and the same thing with me.
In mild conditions that allow for low scoring, Chris Wood of England already is 6 under for the day through 11 holes and is at 4-under par, just two shots off the lead. Given his downfall over the decade since he won his last one, the 2008 USA, they now seem near impossible.
"I told them I tried, and I said "hopefully you're proud of your pops for trying as hard as I did", he said. I know he is going to work on it and make it better, but he still has the PGA Championship.
"Anyone can look at it however they want to, but I'm going to look at it as a positive moving forward and try to learn how to handle the situations a little better next time".