Shortly after the final group started the back nine, six players were tied for the lead, including fellow Americans Kevin Kisner and Kevin Chappell.
Molinari ended a compelling four-day drama on eight-under 276 - the lowest combined score in the eight Carnoustie Opens.
The 14-times major victor had to settle for a closing 71 and a 279 aggregate.
And on the final hole, he hit an approach he judged perfectly to about 10 feet pin-high.
Solid off the tee, suspect at times on the green, Molinari first showed his potential in a World Golf Championship in Shanghai in 2010 when Lee Westwood made his debut at No. 1 in the world. The fan that interrupted an important Tiger Woods backswing late in his round Sunday is potentially far worse.
Tiger Woods led briefly Sunday, but he faded and finished tied for sixth.
However, a brace of bogeys, including one on the 17th, saw him slip to -4 for the day. It's his third win in his last five worldwide starts.
After 37 groups had finished their tournaments and returned to the clubhouse, just three were left out on the course.
The good news for the chasing pack aiming to overhaul joint leaders Spieth and Schauffele was that six of the seven Open winners at Carnoustie have come from off the pace, Ben Hogan the exception after sharing the 54-hole lead in 1953.
Johnson and Kisner were at 6-under 136, one shot ahead of Tommy Fleetwood (65), Pat Perez (68) and Xander Schauffele (66).
However, two holes later he dropped a shot after three-putting at the short 16th, and a rueful shake of the head confirmed his disapproval.
12 and 13 were two really big par putts.
Cohen recording of pay-off talk broadcast on US TV
But, Giuliani went on to say that Trump told Cohen if he were to make a payment to use a check instead of cash. The audio is muffled, but can be heard saying to "pay with cash" and Cohen immediately says, "no, no".
Huge crowds descended on the seaside links and the majority were willing Woods to claim a fairytale 15th major a decade after his last one.
McIlroy nailed an outstanding eagle putt from 40 feet at the 14th to create a six-way tie for the lead.
The name Tiger Woods was back on the top of a leaderboard in the closing stages of a major as the 147th Open Championship headed for a barely believable climax on Sunday. A day after making the cut with nothing to spare thanks to a birdie at the 36th hole, Rose made the most of his reprieve on a serenely calm morning on the east coast of Scotland. He shot 66 in the third round to make a move up the leaderboard.
The American led at seven under before a double bogey on 11 stalled his bid as Francesco Molinari went on to win.
In fact, there are dropped shots everywhere in the final round at Carnoustie, where the wind is playing havoc. "There were some opportunities that slipped by on 2, 3, 4, 5, I felt, but the 6th was the first birdie I've made on a par five all week".
With pandemonium breaking out across the course as 14-times major victor Woods rolled back the years, even Spieth found himself getting sucked in.
He held the outright lead after Spieth bogeyed No. 5 and Schauffele carded two bogeys and a double.
Leaders are on the range at the British Open, where they will be greeted by warm breezes in the final round at Carnoustie.
Of the five players separated by three shots, only Spieth has experience winning a major.
A bogey on the 16th halted the charge but Woods enjoyed a massive slice of luck on the 18th, where his tee shot seemed destined to find the Barry Burn only to take a fortunate bounce to the right to stay on dry land.
Earlier, Justin Rose set the tone with the lowest score in Carnoustie's British Open history, a flawless 64. 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. He shot 76-72 to miss the cut by three shots.
Jordan Spieth was tied for the lead at Carnoustie as he tries to become the first repeat victor in 10 years at the British Open. "I played so well today I don't feel like I should have shot four under".