A joint-low round-of-the-day 66 lifted Kevin Kisner within three of the lead, as overnight leader Bryson DeChambeau slipped down the leaderboard after stuttering to a four-over 75.
And Poulter rekindled that form at the Harbour Town Golf Links after rolling in a 15-foot putt for birdie on the opening hole, before three birdies in four holes on the back nine.
Kraft did his best to battle back, making two birdies and just one bogey on his final 13 holes, but it wasn't enough to make what would have been his sixth consecutive cut, which would have marked the longest streak of his career.
He also sunk a knee-knocking, 12-footer for par on the 18th hole after his approach landed in a front-side bunker.
"Obviously my iron play is incredible, and when I get going I can get hot", DeChambeau told reporters.
"It cost me the cut, most likely", Kraft said.
Englishman Poulter, who won at Houston earlier this month, went out with the early starters and made seven birdies in a bogey-free trip around the course.
The 42-year-old briefly took the lead with a 16-foot birdie on the par 3 seventh. Now, he's one more solid round away from adding another title to his dramatic win at the Houston Open on April 1. 'I can free myself up in my mind. He was imposed a two-shot penalty for touching sand with his hand in a bunker while thinking he was on the greenside fringe. And when you're in that position, sometimes the game feels easy when sometimes it isn't'.
'This might be the first time that I've played so well with a triple bogey, ' Kim said.
Poulter moved out on his at own at 13 under by posting a six-footer at the 13th hole and closed out a blemish-free card along the closing stretch, with List firing a hat-trick of birdies from the 14th to ensure he will go out in the final group on Sunday. 'But I'm having great feel around the greens and I feel really confident'.
"I was just happy in order to play today, &rdquo".