Carter beat two-time Masters champion Higgins 6-3 in a match that lasted nearly four hours and did not finish until 11.29pm.
The 40-year-old is only playing at the Alexandra Palace because Ronnie O'Sullivan did not take up his place in the tournament, yet he has made his first Masters semi-final at the 12th attempt.
— World Snooker Tour (@WeAreWST) January 16, 2020
"It's just great to have the opportunity to be here," Carter told Eurosport after his marathon quarter-final win.
"It was a real scrappy game and I struggled to get into my stride.
"I was aware early on that John was struggling and sometimes you put yourself under pressure to kick on.
"But I just kept running out of position all the time. It gets a bit embarrassing, but sometimes you've just got to suffer it."
Carter won the opening two frames, both of which took nearly 30 minutes to complete, before Higgins responded with a decisive break of 54 in the third.
But Carter took the two frames either side of the interval – Higgins counting the cost of a tricky missed brown in the fifth.
Higgins' struggles ended dramatically as a 140 clearance – the best break of the tournament – and a fluid 73 cut the gap to one.
Carter, however, then won an error-strewn 47-minute frame before closing out victory with a 72 break, by far his best of the match, after Higgins failed to cut a pink into the middle pocket.
Murphy was 3-1 up at the interval and then took a 5-2 lead after Perry missed a red which would have seen him cut the deficit back to just a single frame.
Perry briefly kept his hopes alive of becoming the oldest ever Masters winner.
But Murphy sealed victory with a brilliant pot with the rest, a delicate cannon, and a 32 break.
"It was a big game for me, first chance to get in the semis in five years here," said Murphy.
"Obviously the most important stat is getting to six first, which I did, but there were times in the match I was really struggling.
"It felt it got a bit complicated at the end, my mistakes put me in a lot of trouble.
"On another day I might not be sat here."