He closed with a 2-under 69, playing the final 37 holes on the toughest links in golf without a bogey.
Spieth, already a three-time major champion and still not 25, flexed his muscle on the opening tee shot and didn't let up.
"I did everything the way I thought I needed to do it to win the championship", he added.
Woods parred the 17th but still stood two shots off Molinari as they went to the final hole.
Woods also birdied No. 14 - after playing a pitch shot from the back of the green that is shared with No. 4 - to move back to 5 under, two off Molinari.
The No. 72-ranked English golfer hadn't been hopeful ahead of Sunday's round, saying "today was, really, I wouldn't say a write-off, but I didn't feel I was in the golf tournament".
But his challenge was derailed at the 11th, where his errant drive found the rough and he pulled an ambitious second into the gallery to the left of the green, and he was apologetic as he handed over a signed glove to a bruised spectator. Woods then drained a 20-footer for birdie at the fourth to get within three.
The 42-year-old amazed the golfing world in the final round as he played flawless golf in his first 10 holes Sunday at a hard and windy Carnoustie to move from four back at 5 under into the solo lead at 7 under.
Tiger Woods is in the outright lead of the British Open after a stunning turnaround at Carnoustie.
And then, he grabbed a share of the lead when Schauffele bogeyed the sixth and Spieth doubled. Spieth hit into a gorse bush and made double bogey on the par-5 sixth.
The 42-year-old, who won the last of his 14 Majors a decade ago, has rolled back the clock at Carnoustie and after a sizzling third-round 66 he was at it again.
Tiger with a par on 7 but picked up a stroke and now just sits one shot back of the lead.
Schauffele had to settle for second place with Rory McIlroy, Justin Rose and Kevin Kisner, all of whom had plenty of reason to hope on a warm, windy afternoon that put plenty of bite back into Carnoustie. He finished tied for third.
But it wasn't enough to shake loose the rising Xander Schauffele (67) and stoic Kevin Kisner (68) who will start the final round on level terms with their compatriot at nine under.
Woods had an adventurous 10th hole where he nearly gave away his newfound leader status as soon as he earned it.