Woods missed a great chance to advance further on the 12th, and his pulled drive at 13 took a fortunate kick off the branches and back into play, which he took full advantage of with a birdie four before a neat up-and-down at 15 got him to 10 under.
Molinari, the first Italian to win a major with his victory at last year's British Open, will start in pole position to claim a Green Jacket to go along with his Claret Jug.
"The last two major championships I've been right there", Woods said.
Woods was in the clubhouse with a 5-under 67, his best round of the tournament.
Yet Woods provided the day's biggest talking point, as he surged into contention having survived a dramatic scare on the 14th hole when a security guard - running in a bid to control spectators - slipped and nearly knocked golf's biggest star off his feet.
"It is what it is", Woods said.
After an ordinary outward nine featuring three birdies offset by a pair of bogeys, Woods embarked on one of his trademark charges, carding three birdies after the turn to shoot up the leaderboard.
Woods missed an eight-foot birdie putt to par 17 and missed a 14-footer for birdie at 18.
"Accidents happen, we move on", Woods said of the incident, confirming he suffered no major damage from the guard's boot.
A bold tee shot at the 16th resulted in his sixth birdie of the day, and he parred in to stay in a share of second which ensured him his place in Sunday's final threesome with Molinari and his Ryder Cup team-mate Finau.
As rain lashed Augusta, severe weather forced a 40-minute delay, after which Scott returned to fire a 231-yard second shot onto the green at the par-five hole.
"It was a solid day off the tee, hit the ball well", Molinari said. For none of them, ultimately, go gentle into that good night.
The man - yet to be identified - did hit Woods on his right foot, leading the 14-time major victor to jump in the air several times before limping back to the fairway.
But since then he has shown more than a glimmer of his once-spellbinding form, and has placed himself firmly in contention with a commanding 68 in his second round to remain a shot behind the leaders at 6-under par.