"What a week", Molinari said.
While McIlroy was performing minor miracles, Woods dropped another shot at the 12th as his challenge faded.
Jordan Spieth will take a share of the lead into the final round of the 147th 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.
The 14-times major victor had to settle for a closing 71 and a 279 aggregate.
The Adidas athlete was last year's Rookie of the Year on the tour and made a statement today, finishing just behind Molinari.
Schauffele did not hear the cheers that greeted Molinari's final putt.
Molinari would be the first Italian to win a major championship.
The last time Woods led or held a share of the lead at the end of a round at a major was in 2012, at the PGA Championship.
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.
Kisner and Kevin Chappell both had makeable birdie putts to catch Molinari on the 16th, but both missed and wound tied for second and sixth place. He had sprayed his approach and, perhaps distracted by a crying baby before his chip, the subsequent bogey left him needing a miracle eagle at the last to force a playoff. "I was so frustrated yesterday (after shooting 71) that today was really, I wouldn't say a write-off, but I didn't feel I was in the golf tournament".
Despite a spectacular surge, Woods' run at the trophy and his 15th major championship was spoiled by a double bogey on No. 11 and bogey on 12. He nearly holed the eagle putt from across the green before making birdie.
Tour de France stopped after protesters roll hay into the riders path
Police were said to be arresting some protesters after farmers threw hay bales across the road before the peloton's arrival. Eurosport is the home of the Tour de France . "I want to say that I'm happy to be here after that tough moment", he said.
Trump fires off explosive threat to Iran's leader
Tensions between the US and Iran have been heightened for years, dating back well before the Trump administration. In May, the US left a deal which reduced Iran's nuclear activities in return for the lifting of global sanctions.
Survivors recount deadly Missouri duck boat sinking
The duck boat contained life jackets, but passengers weren't required to wear them, according to Stone County Sheriff Doug Rader . A sunken duck tourist boat that led to the deaths of 17 people faced near-hurricane-like conditions on Table Rock Lake Thursday .
"This wasn't a fluke", 2017 Open champion Jordan Spieth told reporters.
However, Tommy Fleetwood's chances of becoming the first English victor since Nick Faldo in 1992 looked to have gone after he pulled his third shot on the sixth out of bounds, the resulting double bogey leaving him six off the pace.
And then he lost it with two bad holes.
He was three shots behind the leaders, who had yet to tee off in the afternoon at Carnoustie. Woods was leading by one at 7 under.
Just nine shots separated the 79 players who made the halfway cut at Carnoustie, where Paul Lawrie famously came from 10 strokes back with just 18 holes to play to win the Claret Jug in 1999.
In scenes reminiscent of those in the prime of his career, waves of excitement rippled through the massive gallery as 14-times Major champion Woods charged up the leaderboard.
In the midst of such great theater, Molinari made his biggest move by not going anywhere at all.
The 42-year-old, who had five prescription drugs in his system, later pleaded guilty to reckless driving. After a layup, he hit a terrific third shot to about three feet and tapped in for par, capping a third-round 66.
The lead at the top changed several times with 14-time major victor Tiger Woods briefly out in front on his own and 2014 Open champion Rory McIlroy also challenging. Then he missed a 4-footer for par at No. 3. He earned $1.89 million and posted the lowest final score in the eight Opens played at Carnoustie. But Woods rolled in a 40-foot birdie putt on No. 9, followed with short birdie putts on the next two holes and with a two-putt birdie on the par-5 14th, he put his name atop the leaderboard.
Schauffele will be in the last pairing with Spieth after rolling in a long birdie putt on No. 18 for a 4-under 67.