Molinari's opening round of 70 had garnered little attention with Tiger Woods shooting the same score, the grand-slam chasing Rory McIlroy struggling to a 73 and Brooks Koepka and Bryson DeChambeau producing late fireworks to shoot 66.
But the Open champion was among the biggest movers on Friday to quietly climb into contention for a second major title, birdies on the third, eighth and ninth taking the Italian to the turn in 33.
Poulter had needed three more shots to cover the front nine but joined Molinari on five under with a birdie on the 12th and that was good enough to lie just a shot off the lead held by DeChambeau, who had gone to the turn in level par.
Koepka, who is seeking a remarkable fourth major title in his last seven starts, had made an ideal start with a birdie on the first, but then paid the price for a risky escape attempt from the trees on the second and ran up a double-bogey seven.
The 28-year-old recorded just two pars in his first eight holes but remained in the hunt on five under alongside three-time Masters champion Phil Mickelson, who at 48 is trying to become the oldest ever major champion.
McIlroy had struggled to an opening 73 on Thursday, getting under par for the first time with birdies on the 13th, 15th and 16th, only to bogey the last two holes.
That left the Northern Irishman in a tie for 44th place overnight and needing to defy the odds to win a first green jacket to complete the career grand slam as the last 13 Masters winners all enjoyed a place in the top 10 after the first round.
McIlroy headed straight to the putting green following his round after struggling to read putts on the slower surfaces, saying: "I made five birdies, that wasn't the problem.
"I just made too many mistakes.
"And I'm making mistakes from pretty simple positions, just off the side of the green on 17 and 18 being prime examples of that."