Without a bogey in his last two rounds on the windswept Scottish links, Molinari was a worthy victor who arrived at Carnoustie with two victories and two seconds in his past five starts this season.
McIlroy and Rose were in good spirits after finishing tied second, two shots behind Molinari. Molinari didn't make a bogey on his way to a 69 while playing alongside Woods.
McIlroy remained without a major victory since winning his fourth at the PGA Championship in 2014.
Tiger Woods revealed he would likely talk through his Open Championship disappointment with tennis legend Serena Williams.
Donning his traditional blood-red shirt Sunday, Woods needed only six holes to erase his five-shot deficit. "If it misses it, I hit the green and have a birdie putt". I really felt like I had control of the golf ball today.
Pepperell teed off at 11.40 a.m. local time with Phil Mickelson.
For a few fleeting moments Sunday, Tiger Woods - yes, Tiger Woods - led The Open at Carnoustie.
The former Ryder Cup player started the day back in the pack but he galloped his way through the field with three front-nine birdies before gaining more strokes at the 10th, 14th and 16th.
Molinari clinched it with a driver that skirted the edge of a pot bunker, leaving him a lob wedge from 112 yards to 5 feet. Jordan Spieth cost himself by failing to make a single birdie. Spieth hit into a gorse bush and made double bogey on the par-5 sixth.
"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.
After this unbelievable performance, after Tiger Woods almost won The Open, are there really any left? Woods' approach shot stopped 10 feet of the hole.
When Woods proclaimed that he had a shot to win The Open Championship after being within striking distance of the lead, it was a little bit tougher to buy in. So it was pretty emotional there as they gave me some big hugs. And Spieth is trying to become the first repeat champion in 10 years.
Rory McIlroy was bogeying two of his first five holes.
Schauffele remained on course at seven under but he bogeyed at the 17th to put Molinari's lead to two.
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 U.S. countrymen Tony Finau (71) and last year's runner-up Matt Kuchar (72).
Steady opening rounds of 70 and 72 took him into the weekend, before a stunning 65 on Saturday propelled him into contention.
Tiger Woods led the freakin' Open Championship with nine holes to play. He followed up a 64 on Saturday - the lowest round of the week - with a 69 on Sunday to post the first realistic clubhouse target at 6 under par.
Rose also birdied the last to finish on six under, a score that was matched by McIlroy shortly afterwards with pars on the final four holes.
The pair's 72-hole total of 282 was good enough for top-20 finishes, but not good enough to contend for the Claret Jug.
27 - Number of players who finished under par for the week, up from 2007 at Carnoustie (19) and way up from 1999 (0).
It used to be that Woods was steely and superhuman, and no one dared get in his way.
For Woods, Saturdays have been the easiest part of his latest comeback.
With pandemonium breaking out across the course as 14-times major victor Woods rolled back the years, even Spieth found himself getting sucked in.
- Heat expect Carmelo Anthony to sign with the Rockets
- Russian Federation tells USA to release alleged spy Maria Butina after 'fabricated charges'
- Trump and Putin discussed possible referendum in eastern Ukraine
- European Union reaction to United Kingdom white paper 'lukewarm' - Simon Coveney
- EU ruling against Google opens 'opportunity,' rival Mozilla says
- Joss Whedon Announces BUFFY THE VAMPIRE SLAYER Reboot
- Woods shoots 66, in contention at British Open
- 3 on top as Tiger lurks
- How Hillary Clinton Says Russia Could Hack This Year's Election
- Jordan Spieth hoping for hair-raising day with Tiger Woods after close shave