In conditions that were hot but not extreme, the Scot struggled badly with cramp during the last two sets but survived to win 6-3, 7-6 (8/6), 1-6, 7-5.
Haase, who was also troubled physically, served for the fourth set but could not take advantage of Murray's discomfort.
It was another dramatic match on Louis Armstrong Stadium for Murray, who has endured some of his least comfortable moments in grand slams on Flushing Meadows' second court.
The 27-year-old, who next meets German qualifier Matthias Bachinger, said: "I started cramping at the beginning of the third set.
"It was not particularly comfortable. I tried to hang around as best I could and managed to get through.
"I need to work out why that happened today because it shouldn't happen after an hour and 45 minutes, even though it was hot."
The first two sets were competitive and up and down from Murray but there was nothing to indicate the drama to come.
Indeed, the Dunblane man could not have made a better start, racing into a 4-0 lead as he kept Haase firmly on the run.
The one thing that was not working well was Murray's serve and the Dutchman took advantage to retrieve one break, and Murray had to save another break point on his way to taking the set.
Murray is seeded only eighth this year, his lowest in New York since before he reached his first grand slam final here in 2008.
It was Haase who made the better start to the second set, moving into a 3-1 lead as Murray's forehand went walkabout, but the Scot hit straight back.
Murray missed a chance to break for 5-4 but held serve confidently enough to take the set into a tie-break, where he established a 4-1 lead.
He had two set points at 6-4 but could not win either, Haase attempting to whip up the crowd as he saved the second with a vicious forehand.
But Murray found the line to bring up a third set point and took it 8-6 when Haase overcooked a forehand.
Haase had provided one of Murray's uncomfortable Armstrong moments in 2011 when the British number one had to fight back from two sets down in the second round.
And this contest suddenly became a struggle in the third set as Murray, having won the opening game, promptly lost six in a row, barely moving for many balls.
The temperature was soaring and Murray implored the umpire to call for ice.
He came out for the fourth set with the clear intention of trying to make points short but it did not work as Haase made it nine games in a row to move into a 3-0 lead.
Ice was eventually brought, which Murray placed on his left thigh, and he finally stopped the rot to make it 3-1.
He still looked very uncomfortable but it was Haase who called for the trainer after the fifth game for treatment to his foot.
Murray was having some success on his opponent's serve despite his troubles and three times brought up break point in the seventh game.
The return winner he hit to create the last of those opportunities brought on cramp all down Murray's right side but he took the next point with an aggressive return.
Murray did not sit down at the changeover in an attempt to keep the cramp at bay but he dropped serve once again to leave Haase serving to make it two sets all.
The Dutchman made a mess of it, though, handing Murray the break back with a double fault, and two games later he brought up the chance to serve for the match with a forehand thumped onto the sideline.
Still it was not straightforward, Murray having to save three break points before finally clinching victory after a gruelling three hours and eight minutes.