Woods and Molinari will join Tony Finau, who also sits at 11-under, to form the final threesome for Sunday morning's final round - which was adjusted and moved up due to impending severe weather.
After bogeying the fifth hole for the third consecutive day, Woods appeared on the verge of getting lapped, but instead made three consecutive birdies before the turn.
Molinari's rise, under the long-time swing tutelage of Denis Pugh, has coincided with his decision three years ago to work with performance mentor Dave Alred, the guru behind England's World Cup-winning rugby star Jonny Wilkinson and who also helped steer Luke Donald to golf's world No.1 slot.
"All parts of my game I'm very confident in at the moment, and you know, when I am hitting it well, that's where I play my best golf", Scott said.
Woods - showing no ill effects from yesterday's inadvertent interaction with a security guard - looked good as new, even for a 43-year-old who hasn't won a major in almost 11 years.
The 2012 U.S. Open champion stumbled on the sixth hole, committing a "sloppy" bogey, but he recovered spectacularly by logging seven birdies over the next dozen holes.
After a nice two-putt for par at the 16th, Woods had an 8-footer for birdie at the 17th and a 14-footer at the 18th, both of which did not drop.
"I just let the round build", he said. He hit the 2-iron instead, and the gamble paid off when the ball ended up 5 feet from the pin.
"I'm aware that it's not going to be easy tomorrow [Sunday]", he added.
Scott's 69-68 marked the first time he's broken 70 in his first two rounds at the Masters.
"I'm fine. It's all good", Woods said.
Rose suffered a cruel lip-out with a seven-foot putt at the 17th hole and then went from bad to worse, hardly touching the hole with a four-footer at the last.
Tiger Woods made the largest logjam in Masters history look even bigger by making Augusta National sound as loud as ever.
"As a kid I always wanted to compete against him", said 29-year-old Finau of Woods. He tacked on birdies on 13, 15 and 16 to play his way into the final group.
When Molinari won the 2018 Open Championship, his first major, he was paired with Woods on Sunday at Carnoustie. There are so many great players in with a chance.
"I think tomorrow is a day when you just really want to produce some good shots, not try too much and play yourself out of the tournament".
Woods, the victor of 14 majors, and Tony Finau will be alongside Molinari when round four begins earlier than scheduled, with organisers hoping to finish the event before an anticipated barrage of poor weather.