Woods led at Carnoustie with eight holes to play before finishing sixth behind playing partner Molinari, who held his nerve to card a bogey-free 69 to become the first Italian player to win a major title.
The 36-year-old Ryder Cup star takes a two-shot lead over Woods and Tony Finau into Sunday's final round at Augusta, with the tee times brought forward in an effort to avoid thunderstorms forecast to hit the course in late afternoon.
"I think, to be honest, every tournament is different and every time is a different story," Molinari said after a superb third round of 66. "He (Woods) obviously loves this place and he's playing great golf.
"So I'm aware that it's not going to be easy and I can just do my best. But it's not like I can only worry about him. There's a lot of guys I think in with a chance.
"We've seen in the past years that a few shots' lead really doesn't mean too much and we've seen today that you can shoot seven, eight under the way the course is playing. I think there's a lot more guys with a chance.
"Coming from Italy, it's exciting just to walk down Magnolia Lane. I don't have to be in the last group on Sunday to be excited. It's a special week. It's always been a special week for me, even in the last few years where I didn't play so well.
"I'll just try to enjoy tomorrow as much as possible, and again, do my best, shot by shot, don't get ahead of myself and see what happens."
All times BST
From first tee:
12:30: Aaron Wise, Marc Leishman (Aus), Kim Si-woo (Kor)
12:41: Matthew Fitzpatrick (Eng), Kevin Tway, Patton Kizzire
12:52: Henrik Stenson (Swe), Hideki Matsuyama (Jpn), Cameron Smith (Aus)
13:03: Jordan Spieth, Tommy Fleetwood (Eng), Corey Conners (Can)
13:14: Jason Day (Aus), Bubba Watson, Lucas Bjerregaard (Den)
13:25: Justin Thomas, Phil Mickelson, Jon Rahm (Spa)
13:36: Adam Scott (Aus), Patrick Cantlay, Thorbjorn Olesen (Den)
13:47: Louis Oosthuizen (SA), Dustin Johnson, Rickie Fowler
13:58: Matt Kuchar, Justin Harding (SA), Xander Schauffele
14:09: Brooks Koepka (USA), Webb Simpson (USA), Ian Poulter (ENG)
14:20: Francesco Molinari (Ita), Tony Finau, Tiger Woods
From 10th tee:
12:30: Kyle Stanley, (a) Viktor Hovland (Nor), Charley Hoffman
12:41: Kevin Kisner, Bryson DeChambeau, (a) Takumi Kanaya (Jpn)
12:52: Rory McIlroy (NI), Keegan Bradley, Gary Woodland
13:03: Jimmy Walker, J.B. Holmes, Charles Howell
13:14: Kiradech Aphibarnrat (Tha), Kevin Na, (a) Alvaro Ortiz (Mex)
13:25: Patrick Reed, (a) Devon Bling, Keith Mitchell
13:36: Tyrrell Hatton, Satoshi Kodaira (Jpn), Andrew Landry
13:47: Bernhard Langer (Ger), Rafael Cabrera Bello (Spa), Eddie Pepperell (Eng)
13:58: Branden Grace (SA), Martin Kaymer (Ger), Li Haotong (Chn)
14:09: Emiliano Grillo (Arg), Zach Johnson, Billy Horschel
14:20: Trevor Immelman (SA), Alex Noren (Swe)