Spieth, who started the day in a three-way share of the lead with Kisner and Shauffele, crashed with a 76 to wind up equal ninth a further stroke back with USA countrymen Tony Finau (71) and last year's runner-up Matt Kuchar (72).
Molinari made history by becoming Italy's first ever major victor and admitted he would probably need to change his flight home so he can celebrate becoming the first player from the country to win a major.
He got one back on the par-five 14th but Molinari also birdied to take him two shots ahead of the American.
That was his best round of the week, highlighting that scoring opportunities were on offer for those who hit their stride. He made double-bogey on the 11th after needing four shots to reach the green.
Molinari, whose older brother Edoardo won the 2005 U.S. Amateur and played with him in the 2010 Ryder Cup, comes from a country known more for soccer and skiing than golf.
"Who hasn't?" Spieth said after seizing upon a calm Carnoustie for a 6-under 65 to tie for the lead with Kevin Kisner and Xander Schauffele. He opened with 13 pars, including the 12th and 13th holes as Woods lost the lead with a double bogey and a bogey. Spieth said. "I've played it out at Augusta in my head". Once they saw guys going low, it was nice to get a following out there.
A day in which everything seemed to be coming up trumps for Woods took a turn for the worse on the 11th.
Holes were running out and the par-three 16th which had withstood all the preceding golfers during the round also refused to yield to him.
His tee shot went into the crowd and, despite a fortunate bounce, his next shot over a bunker went off the back of the green.
"Serena will probably call me and talk to me about it because you've got to put things in perspective", said Woods, who eventually finished tied for sixth, three shots behind playing partner and champion Francesco Molinari.More news: Trader Joe's Suspected Gunman In Custody, Hours After Taking Hostages — LAPD
Woods made a 40-footer on the ninth for his third birdie against no bogeys on the front nine to get to 3-under par for the tournament.
The Northern Irishman's clenched-fist celebration underlined his delight and, no doubt, sense of belief.
Molinari didn't drop a shot over the weekend at Carnoustie.
Justin Rose, who made the cut on the number with a birdie on his final hole, matched the Carnoustie record for the Open with a 64. So when he rolled in a 40-foot birdie putt on number nine, stiffed an approach inside two feet for birdie at number 10, and almost drove onto the green to set up a tap-in birdie at number 11, there was a feeling something special was happening.
Woods, who had six birdies in his round of 66, is another stroke back alongside Rory McIlroy (70), Webb Simpson (67), Alex Noren (67), Matt Kuchar (70), Tommy Fleetwood (71) and Zach Johnson (72).
When he turned to see the scoreboard above one of the packed grandstands, Schauffele's name had joined Molinari's at the top. He parred the final three holes, missing a 6-foot putt on the final hole that would have given him his seventh runner-up in a major.
With the stretch having been so hard throughout, the victor would really have to earn their glory.
Molinari went one better, his second setting up an easy birdie and laying down a marker at eight under for the four players left on course. Schauffele is 25. Spieth will celebrate his 25th birthday next week.
England's Eddie Pepperell had set the early clubhouse target after a superb 67 left him five under par, four shots off the lead.
Spieth and Schauffele have parred their first two holes.