Warren defeated world number 12 JB Holmes in his opening group match on Wednesday and knew victory over Brooks Koepka on Thursday, coupled with Holmes beating Russell Henley, would see him advance.
The 34-year-old from Glasgow looked certain to fulfil his side of the bargain when he birdied the 15th to move three up with three to play, only to bogey the 16th and 17th to hand Koepka a lifeline.
Warren looked to have done enough to seal victory when he found the green in two on the par-five 18th, with Koepka in a greenside bunker.
However, Warren then three-putted – missing from two feet for birdie – and Koepka, who had almost holed his bunker shot, safely converted his birdie putt to take the match into extra holes.
Warren again had the upper hand when he found the green on the par-five first in two and Koepka found more sand with his approach, but Koepka got up and down to match Warren’s two-putt birdie and then birdied the next to seal a remarkable win.
That left Warren needing to beat Henley on Friday and hope Holmes gets the better of Koepka, which would result in a play-off with Warren, Koepka and Holmes – who beat Henley on the 19th hole – all having a 2-1 record.
Former US PGA champion Keegan Bradley had been the first player eliminated when he lost his second match of the week to Bubba Watson, who will face Louis Oosthuizen in a decider on Friday, the man he beat in a play-off for his first Masters title in 2012.
“We are not playing for a green jacket but I need to play well so I need to have a good start and make birdies,” said Oosthuizen, who beat Miguel Angel Jimenez to eliminate the Spanish veteran.
Graeme McDowell had also been eliminated after losing to Harris English, with Rickie Fowler beating Shane Lowry in the other group 13 match thanks to holing a string of lengthy putts on the closing stretch.
“I feel bad to do that to him but it was nice to finally make some putts,” Fowler told Sky Sports 4.
“It would have been nice if we were throwing birdies at each other but my back was up against the wall and I needed to make those putts for par to keep the match going.”
Lowry, who had been two up with six to play, wrote on Twitter: “Well all I can say about that is… Wow!!! Fair play to @RickieFowlerPGA. Really enjoyed the battle out there today. #lovematchplay.”