The 14-time major champion began the day four shots back from the trio of overnight leaders, headlined by defending champion Jordan Spieth. Paired with Woods throughout his final round at Carnoustie, Molinari parred his first 13 holes before making a birdie at the par-5 14th hole.
Jordan Spieth, Xander Schauffele and Kevin Kisner might be well-advised to avoid reading up on their Carnoustie history as they prepare to battle for the Claret Jug on Sunday.
Woods might not win The Open this week, and he might not win the PGA Championship at Bellerive in August, and we can still debate whether or not he can catch Jack Nicklaus' record of 18 major championships.
Fresh from his best major round in seven years on Saturday, with the 42-year-old signing for a 66 on Saturday, Woods surged to seven under, with playing partner Francesco Molinari one further back.
The Englishman, who had only edged his way past the cut, sunk seven birdies to go within two shots of the leaders at four under. Speaking to reporters Saturday, Woods said he's happy with how he's playing.
The 37-year-old said: "Momentum wise, birdieing the final hole picking up where I left off last night got me into the round early". In theory, he could still win a place in next year's field at Royal Portrush by coming through qualifying or winning next week's Senior Open Championship, but the Scot appeared to accept that it was very unlikely that he would...
"I could have cleaned up the round a little bit", he said. I just needed one more birdie before the sixth hole. But if anybody, at this point, is still saying Woods won't win another major, they are just guessing, because clearly, Tiger can win a 15th major. Spieth hit 8-iron through a gap in the trees for a birdie-birdie start to the back nine, and he dropped only one shot - not four like he did on Thursday - over the four closing holes at Carnoustie for a 67. Molinari got into contention with a Saturday 65 (he entered the final round three back) and then bought it home Sunday.
Another American Kevin Chappell is two shots off the lead with Italy's Francesco Molinari on six under and Woods another stroke back alongside Rory McIlroy, Tommy Fleetwood, Webb Simpson, Alex Noren, Matt Kuchar and Zach Johnson.
American Luke List and 2016 Open champion Henrik Stenson both carded 69s while Denmark's Thorbjorn Olesen was the first man to make an impression on the leaderboard as he birdied two of his first three holes to reach five-under-par.
"I want them to see Dad do what he's done for most of his life", he told NBC, "and make them feel and watch what their father can do".
"It's all going to be based off the wind, how much the wind blows", he said after finishing with a one-under 70.
Rose's round matched the lowest score shot in an Open Championship at Carnoustie, a traditionally hard course on Scotland's east coast.