The Scotsman missed a black off the spot when he seemed certain to complete the win allowing 10th seed Allen to make a decisive 44 to secure his place in the last eight.
Higgins felt he got a kick on the crucial ball at the wrong moment, but admitted his downfall was down to his own carelessness.
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"It seems to happen to me a lot in my career," said Higgins. "That's the miss that cost me the game. If you don't pot balls at the key moments, you don't win snooker matches."
The Northern Irishman Allen - whose best at the Masters is two semi-final appearances in 2011 and 2015 - will face Marco Fu or Judd Trump in the quarter-finals on Thursday evening.
"If you don't play well, you are going home against the best players in the world, " said Allen.
"I like to think I can raise my game against the best players in the world, but that's no use in our sport, you have to play well against everybody.
John Higgins at the Masters - picture Tai Chengzhe/World Snooker.
Image credit: Eurosport
"It was very tense in the last frame. He was shaking a bit in the last frame. It shows how much it means to John that he was feeling it at 5-5 in the first round of the Masters.
One thing I've never struggled with in my career is bottle. I'm pretty good when it gets close. Unfortunately, I don't get it close too often.
Higgins led 2-0 and 3-2 courtesy of breaks of 111 and 54 only for Allen to move 4-3 ahead with runs of 81 and 104.
Higgins levelled at 4-4 by dominanting the eighth frame and he pounced on an Allen miss on 67 to force a re-spotted black with his own timely contribution of 67.
The two-times Masters winner had a couple of attempts to double the black before Allen holed a brilliant long pot to move 5-4 clear.
Again Higgins drew level at 5-5 with a rapid 77 before his missed black in the decider finally swung the match in his opponent's favour as Allen held his nerve to leave Higgins needing a couple of snookers.
Despite his best attempts, Allen rolled the pink into a baulk pocket to finalise his progress to the last eight for the fourth time in the past five years.
In Monday night's match, Joe Perry set up a quarter-final with Ding Junhui on Friday evening by completing a resounding 6-1 win over 2015 world champion Stuart Bingham.
"I just didn't feel match-sharp out there," said Bingham, who became a dad for the second time on Saturday night. "It was good to watch Joe play, but obviously I didn't want to be in the chair watching him."
Perry won only his second match in nine outings at the Masters boosted by runs of 116, 77, 76 and 107.
His only previous win at the Masters came two years ago when he beat Ding in the first round.
John Higgins (Sco) 5-6 Mark Allen (NI): 74-29, 111-1 (111), 1-83 (51), 13-68, 74-7 (54), 0-81 (81), 0-112 (104), 70-8, 67-74 (Higgins 67, Allen 67, won on re-spotted black), 77-0 (77) 44-69
Stuart Bingham (Eng) 1-6 Joe Perry (Eng)
6-126 (116), 65-72 (Perry won on re-spotted black), 138-0 (132), 32-77 (77), 32-82 (76), 0-107 (107) 55-69
Desmond Kane at Alexandra Palace
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