Shortly after the final group started the back nine, six players were tied for the lead, including fellow Americans Kevin Kisner and Kevin Chappell.
Jordan Spieth will take a share of the lead into the final round of the 147th British Open Championship as he looks to become the first back-to-back victor for a decade.
He closed with a 2-under 69, playing the final 37 holes on the toughest links in golf without a bogey. It will be interesting to see going forward, but this was his first taste of major championship drama for quite a while.
Schauffele, the PGA Tour rookie of the year last season, holed a 30-foot putt from behind the 18th green for a 67.
Schauffele had to hole out a wedge on the final hole and checked up well short of the hole.
Molinari pocketed 1,625,387 euros ($1,907,183.59) for the win.
There is no question that Tiger Woods is a contender once again and that is all that matters.
Tiger Woods didn't entirely hurt his chances in the British Open.
However, the 42-year-old 14-time major victor had his rivals rattled when he grabbed the lead thanks to two birdies on the front nine while they were dropping shots. It's his third win in his last five worldwide starts. He nearly holed the eagle putt from across the green before making birdie.
Jordan Spieth was wounded physically but not mentally after failing to defend his Open title at Carnoustie on Sunday.
With Schauffele recovering from four dropped shots in three holes on the front nine with birdies at the tenth and 14th to join him on seven under, Molinari got a birdie. Perez was tied for the lead until he hit into a bunker on the 18th hole and took bogey. Rose is No. 3 and almost joined them until he made an 18-foot birdie putt on the last hole to make the cut on the number.
Large crowds are following the 42-year-old Woods, who has yet to win in his latest comeback.
McIlroy had struggled to make any headway but when he eagled the 14th he rocketed to the top of the leaderboard.
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Onlooker Michael Drejka got out of his parked vehicle and began "harassing" her about being parked in a handicap space, she said. Inside the store, a witness told the store clerk there was an argument outside. "It happened to me the first time .
The American topped the leaderboard until his hopes nosedived following a double-bogey at the 11th and another dropped stroke at the next hole.
Woods shot a five under par 66 to leave himself at -5 for the day & tournament.
Spieth and Schauffele have parred their first two holes.
And for the first time, all of them get to experience a final round at a major with Woods in the hunt.
Leaders are on the range at the British Open, where they will be greeted by warm breezes in the final round at Carnoustie. Three other players are within three shots, while Tiger Woods is in a group of eight golfers at 5 under. Strategies changed. Slightly softer conditions meant power players who were going beyond the trouble hit more irons, and shorter players hit more drivers and fairway metals.
The biggest blunder belonged to Woods, his red shirt blazing against the yellow grass of dry Scottish summer.
"I'm a little ticked off at myself for sure", said Woods, whose third-round 66 was his lowest score in The Open since he raised the Claret Jug for the third time at Hoylake in 2006.
Kisner had the 54-hole lead at the PGA Championship past year.
Woods made the most of his good fortune to make par.
He said: "I played well today, I really did".
This is the 16th time he has been in at least a share of the lead in the majors in the five years he has been playing them on a regular basis.
Asked if Spieth felt he and his rivals would now be dealing with Woods's name hovering on leaderboards for a number of years to come, the three-times major victor replied: "I think so, yeah".