The English Ryder Cup star, who must win in Texas this weekend to snare the final invitation on offer for Augusta National, produced seven birdies and no birdies to move to 14 under.
Poulter was joined at the top of the leaderboard by Texas resident Beau Hossler, who drained a birdie on the last to move to 14 under with a 69.
Poulter's performance was the latest twist of a roller coaster week for the 42-year-old, who was informed incorrectly during the WGC–Dell Techologies Match-Play championship in Austin that he had earned a place at Augusta.
He later learned that that information was wrong, just 10 minutes before he teed off in a semi-final with Kevin Kisner, which he duly lost.
Poulter, who delayed his decision to compete in Houston, said he had still been irked by the mix-up when he arrived this week, shooting a 73 in the first round as he attempted to force the pace.
"Maybe I was a bit angry on Thursday, maybe I was kind of forcing, trying to force my way into a tournament," Poulter said.
"It didn't work, had to rethink it, had to kind of blow the cobwebs out Thursday night and reset and go again."
Although one strong round on Sunday could vault him into the Masters, Poulter said he would be relaxed either way.
"I'll have no emotion at all. I'm going to go play golf," he said. "I'm in a no-lose situation. I haven't won a stroke play event on the PGA Tour and I'm in a position where I've got an opportunity to."
Poulter had earlier kick-started his third round with four birdies in his opening eight holes.
He then holed a 13-foot birdie putt at the par-4 10th hole to move to 12-under ahead of a superb tee shot to just seven-feet at the par-3 16th. He then split the fairway at 17 to land his second shot to just six-feet for a seventh birdie and a 65.
Poulter headed to the scorer's hut leading at 14-under par and waited as Hossler, the overnight leader, did his best to match his English rival's effort.
The 23-year-old struggled with just one birdie in his opening 10 holes only to drop a shot at 11 before the Austin resident steadied to birdie the next two holes and move to 13-under par.
He came to the last and duly delivered the birdie to tie Poulter, and much to the wild applause of a partisan Texas crowd.
While both Poulter and Hossler are desperately seeking the win to qualify for Augusta there are four players in third place at 12-under par and if any of those should win they will be heading to the Masters instead.
They include – Australia's Greg Chambers (65), Argentina's Emiliano Grillo (67), American Kevin Tway (69) and Ireland's Paul Dunne (69).
Looming at 11-under par are two players already qualified for the Masters, Sweden's Henrik Stenson and American Matt Kuchar who signed for 69s.
Jordan Spieth is four off the pace at 10 under after a third round 71.