Murray defeated the 70th-ranked Dutchman 6-3, 7-6 (8/6), 1-6, 7-5, but was at a loss to explain the painful cramp that struck without warning -- and could have cost him the match.
"I could have easily lost that match," said Murray, who thought he'd have been in real trouble in a fifth set. "I was very close to losing the match."
The Scot, who hasn't reached a final since lifting the Wimbledon trophy in 2013, staggered through the third set, and twice rallied from a break down in the fourth to finally close it out after just over three hours.
The 27-year-old, who won the US Open in 2012, said he arrived at Flushing Meadows fitter than he has been for any Grand Slam of the year.
But the match against Haase left him looking for answers.
"I don't know exactly why it happened today," said Murray, who trains in the steamy heat of Miami and didn't think the 30 C temperatures on Louis Armstrong Stadium were excessive.
"The conditions in Miami were significantly hotter and more humid than it was out there today," he said. "And at the time it happened I wasn't exhausted. It just happened."
In the end, however, he was safely through, gaining the decisive break in the 11th game of the fourth set then saving three break points in the following game before closing out the match.
With two-time defending champion Serena Williams and five-time champion Roger Federer not playing until Tuesday, reigning Wimbledon champion Novak Djokovic and French Open winner Maria Sharapova headlined the opening night.
Top-seeded Djokovic, recently married and soon to be a father for the first time, is gunning for a fifth straight US Open finals appearance and a second title to add to the one he won in 2011.
He was set for a first career meeting with Diego Schwartzman, a 22-year-old from Argentina who is playing in his second career Grand Slam and the first tour-level hardcourt tournament of his career.
Sharapova, who won her fifth career Grand Slam title at Roland Garros this year, returns to Flushing Meadows after missing last year's US Open with a shoulder injury.
She'll face fellow Russian Maria Kirilenko, a former top-10 player now languishing at 113th after a spate of injuries.
Australian Open winner Stan Wawrinka, seeded third, moved smoothly into the second round with a 6-2, 7-6 (8/6), 7-6 (7/3) victory over Czech Jiri Vesely.
Milos Raonic, the big-serving Canadian who is among the young guns vying to end the Grand Slam hegemony of the game's Big Four -- Djokovic, Federer, Rafael Nadal and Murray -- pelted Japanese qualifier Taro Daniel with 20 aces in a 6-3, 6-2, 7-6 (7/1) victory.
- Normal nerves -
In early women's action, second seed Simona Halep shook off first-round jitters to defeat unranked US wildcard Danielle Collins 6-7 (2/7), 6-1, 6-2.
Agnieszka Radwanska showed no sign of big-tournament nerves, but the fourth-seeded Pole said she felt them nonetheless in a 6-1, 6-0 drubbing of Canadian Sharon Fichman.
"First match is always tricky," said Radwanska, who allowed her 112th-ranked opponent just three points in the second set.
Romania's Halep, who has quietly risen to number two in the world thanks to eight career WTA titles and with a little help from the injury-enforced absence of Li Na, admitted she was a bit intimidated to find herself first up for the tournament on the massive Arthur Ashe Stadium court.
"It's not easy to manage the situation, but I think after first set did I very well," said the 22-year-old who was runner up to Sharapova at Roland Garros. "I played better than first set. I started a little bit nervous, but it's normal."