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Murray and Peers saved a match point to beat Americans Steve Johnson and Sam Querrey 6-4 6-7 (2-7) 7-6 (9-7).
It is the second successive Grand Slam final for Murray and Peers after their run at Wimbledon in July.
They will play French duo Pierre-Hugues Herbert and Nicolas Mahut, who beat Inglot and Robert Lindstedt 7-5 6-2.
"My nerves are shot but we're so happy to win and get to the final," said Murray.
"We played a very high level match and the other guys were serving really well but we managed to turn it around."
Murray and Peers took the opening set but the Americans, who failed to take any of their first four break points, won the second set 7-2 on a tie-break.
Eighth seeds Murray and Peers moved 5-3 ahead in the deciding set and were serving for the match only to be immediately broken back by the Americans.
The third set went to a tie-break, Murray and Peers saving a match point before advancing into the final.
Earlier on Thursday, Inglot failed to make the final, although it was still his best run at a Grand Slam after two quarter-final defeats in the Australian Open.
Inglot and Sweden's Lindstedt were playing in only their fourth tournament together, having won the US Open warm-up event in New Haven, but could not build on an early break and struggled on serve as Herbert and Mahut took control.
Two breaks of serve gave the French duo the first set in 47 minutes, and they moved 3-1 clear in the second on the way to matching their run to the Australian Open final in January.
"Credit to them, they were just better on the day," Inglot said. "It wasn't my best day, I thought I let us down a bit so I'm a bit disappointed with that."
Britain's Jamie Murray and Australian John Peers moved into the US Open doubles final, but Dom Inglot missed out on his first Grand Slam final.