It caps a stunning stretch for the 15th-ranked Molinari, who has won on the PGA and European Tours in the previous two months.
Birdies at the fourth and sixth hoisted Woods to seven-under and left him just a shot behind Spieth and Schauffele in third place after they both came to grief with bogeys on the fifth.
Molinari, 35, is the first Italian golfer to ever win a major championship.
Overnight co-leader Zach Johnson gave up a shot early at the British Open when his drive at the fourth found a hazard, but he bounced back by holing what will nearly certainly be the putt of the week at the par-5 sixth hole.
Tiger Woods is in the outright lead of the British Open after a stunning turnaround at Carnoustie.
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Rory McIlroy has bogeyed two of his first five holes and is now 3 under, six shots off the lead, while Tommy Fleetwood bogeyed No. 5 to give up the stroke he picked up at the first before making double-bogey at No. 6.
Woods has not played in the Open since 2015 due to injury and is without a victory since 2013, while the last of his 14 major titles to date came a decade ago in the US Open. Any time a player has to fire a shot out of a gallery and there are wide-eyed spectators lining a narrow chute back toward the fairway, you're always a bit amazed that someone isn't tattooed with a golf ball. At the long par-three 16th his tee-shot just failed to stay on the green and he three-putted. But while while the wheels fell off for his playing partner, three-time major victor Jordan Spieth, Schauffele held things together and still had a chance to win with two holes to play.
Woods made the most of his good fortune to make par.
It was his only bogey of the day, but the closing holes at Carnoustie are brutal.
Three players - including defending champion Jordan Spieth - are tied at the top of the leaderboard at 9-under par.
Saturday's third round saw 41 players card under-par rounds but 24 hours later it appeared to be a case of hanging on.
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Puigdemont's lawyer, Jaume Alonso-Cuevillas, declared triumph, writing on Twitter: "It looks like we have a memorable summer".
Spieth finally hit the front alone when he birdied the 16, but 2017 US PGA Tour Championship victor Schauffele and smooth-putting Kisner made their combination of great birdies and par saves late to keep pace.
He missed the green with his approach but his recovery game was dialled in and he saved par in style.
Sensing that something special was brewing, huge crowds swarmed across the Scottish links trying to catch a glimpse of the 42-year-old's barnstorming charge.
Jordan Spieth and Xander Schauffele, the last pairing in the final round of the British Open, have teed off at Carnoustie where quite a stiff wind is blowing.
"We've got pretty much a new tournament tomorrow", Spieth said.
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