An out-of-sorts Higgins managed to compile runs of 54, 140 and 73, but it was Carter who took control of the crucial moments to progress to a first Masters semi-final of his career.
Carter will face Shaun Murphy in the last four after he beat Joe Perry earlier in the day.
World No 17 Carter won the opening two frames during a low-quality start before Higgins hit back in a third frame notable for three fouls in succession.
Higgins was the first culprit as he potted the white with Carter nearly needing snookers. Carter quickly repaid the favour, only for Higgins to foul again on the next shot as he inadvertently sent the black into the centre pocket. Higgins would take the frame on the back of a 54 break.
Carter won the next to move 3-1 ahead at the interval.
‘Unbelievable! Unbelievable what’s happening here!' – Foul from Higgins leaves commentators aghast
A 38-minute frame also went Carter’s way immediately after the break as Higgins’ below-par play continued with a missed brown proving costly. However, Higgins hit back with a superb 140 break followed by a 73 to close to within one frame.
Those two quick frames were followed by the longest frame of the match – at 47 minutes – eventually won Carter, and the 40-year-old made a 72 to seal his progression in style.
"It was real scrappy game, and I really struggled to get into my stride" said Carter in the Eurosport studio after the match.
I was aware early on that John was struggling and sometimes you put yourself under pressure and I kept on running out of position but I was pleased with the way I took the last frame.
Earlier, Murphy reached his first Masters semi-final since 2015 after beating Perry 6-3 in the quarter-finals.
The 2005 world champion put together runs of 51, 60, 66 and 120 to progress to the last four, but the 37-year-old told Colin Murray post-match that he felt Perry had been unlucky.
I am obviously pleased to win. It was a big game for me, first chance to get in the semis in five years, and the most important stat is getting to six first, but there were times in the match when I was really struggling and on another day I might not be sat here as I thought Joe was so unlucky, so often.
Murphy, who won the tournament the last time he reached the semis, took the opening frame of the match against world No 16 Perry, but let the second slip from a good position. Leading 64-34, Murphy missed a double to the middle and Perry cleaned up to win the frame on the black.
However, Murphy replied with a break of 66 and then came from behind to take the fourth frame and establish a 3-1 lead.
Shaun Murphy: This is my best chance to win a Triple Crown event
Just as it looked as though Murphy was in control, he missed a red to the middle which gave Perry the chance to cut the deficit.
There was no let up from Murphy in the next frame as a break of 120 saw him re-establish his two-frame lead.
Perry then squandered an opportunity to close again as he missed a long red with a 61-25 lead, and Murphy took advantage to move 5-2 ahead.
Shaun Murphy steals in to open up three-frame advantage
Even though Perry managed to win the next frame, that was the end of his comeback bid as Murphy won 6-3 to set up a semi-final against Ali Carter or John Higgins, who meet in the quarter-finals from 7pm on Thursday evening.