Three of the world's top five players missed the cut, including top-ranked Dustin Johnson and No. 2 Justin Thomas.
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.
Molinari settled for the best cheer of them all.
Woods leaked another shot at the tough 12th and, although he got back to within a shot of the lead with a birdie at 14, he could only par in from there for a 71 which left him three behind champion Francesco Molinari.
"Chaotic is probably the best way to put it", he said.
"I didn't play the wind the right way on those two holes", he added.
Tiger Woods says not being able to convert a one-shot lead at the Open Championship into a shock 15th major title is "going to sting".
Before heading to the course on Saturday, Spieth went into Carnoustie for what he called "a very British haircut", just as Tiger Woods was rolling back the years with his third-round 66 in front of huge galleries.
A massive drive on the following hole nearly took him to the green and he duly went on to birdie that as well to pile the pressure on overnight leaders Kevin Kisner and Zach Johnson. They ran out of holes.
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. Woods was in control of his game at the halfway point, with a one-shot lead at the turn, within reach of a 15th major. "I've had a career's worth in a few years", said Spieth, who admitted he found it "difficult" to hand over the Claret Jug when he arrived in Carnoustie on Monday. They all dropped shots on a day when there was no time to be going backward. Playing with Tiger was another challenge. But I felt really good this morning.
"I think I need more feel in my putting".
The yellow grass of a dry summer in Scotland comes to life Sunday because of a red shirt at the British Open.
Let's be clear: The White House week of walk-backs
He said he still doesn't know what Trump discussed with Putin during the leaders' one-on-one meeting in Helsinki . To be fair to Trump , he could not support a line of thought that he won the election with Russian support.
McIlroy eagled the 14th from 40 feet as he finished 6-under-par, but it was Molinari who stayed solid, stayed calm and got the job done. He looked like the Woods of old, and it felt like old times when everyone he was chasing began to collapse.
Spieth flew his drive right into the crowd and his second shot from the rough with a three-wood ended with his ball in a gorse bush. "Emotionally draining", was his description for the course, the final four holes of which have damaged him every day.
Schauffele bogeyed Nos. 5 and 6, then double-bogeyed the seventh hole.
"I got beaten up around here a few times already in the past", Molinari said.
His anger over his mistakes was tempered by perspective, comparing it to Serena Williams losing in the finals at Wimbledon.
"I saw he played pretty well today". Serena and I are good friends. "So that was the goal to make sure I got in the clubhouse ahead of him".
"I love where my putter is and love my position going into the weekend", Kisner said.
Even a long hug from his two children afterward wasn't enough to ease the sting. His first bogey of the day left him at five-under.
Bhatia, from Wake Forest, North Carolina, missed a tying 4-foot birdie putt on the par-5 35th hole. Another bogey on the next hole, and he was basically out of the tournament.
This round wasn't flashy, and neither is Molinari. A birdie at the sixth followed when he buried a 7-footer.
But not the most memorable. I putted well. I holed out when I needed to.