The British former world number one played in a manner reminiscent of his best as he led by two sets to one.
But Tsitsipas fought back to win 2-6 7-6 (9-7) 3-6 6-3 6-4 in New York.
Dan Evans is the only Briton through after defeats for Heather Watson, Harriet Dart and Cameron Norrie.
Johanna Konta withdrew from her first-round match because of a thigh injury, meaning teenager Emma Raducanu and Katie Boulter – who play on Tuesday – are the nation’s last remaining hopes in the women’s singles.
Murray, like so often in the past, was the British player who stole most of the attention on the opening day of the final Grand Slam of the 2021 season – even in defeat.
Now ranked 112th in the world, the Scot was aiming for his biggest win by ranking since his career was derailed by a serious hip injury that left him facing retirement.
Fired up throughout and relishing the magnitude of the occasion, 34-year-old Murray moved freely and hurt Tsitsipas with his quality from all over the court.
A seismic shock beckoned before 23-year-old Tsitsipas – one of the favourites for the title – maintained his focus and bided his time to win in four hours 49 minutes.
“It did not come easy to turn the match around. I had to make a lot of sacrifices,” Tsitsipas said.
After being irritated by Tsitsipas’ eight-minute bathroom break before the deciding set, Murray exchanged a terse handshake with his opponent before continuing to loudly chunter “it’s a joke” while he gathered his belongings.
Even the affectionate and rousing reception which accompanied the 2012 champion off the court could not hide his annoyance.
Defying the odds is something Murray has managed to do on several occasions in recent years, but beating an opponent of Tsitsipas’ standing over five sets at a Grand Slam would have been a new benchmark.
Niggling injuries since major hip surgery have continued to hamper his progress and, having played only 12 matches this year before the US Open, doubts remained about his chances of ever competing again with the world’s best.
While Murray arrived in New York speaking positively about his physical state, he said it was his game that needed to be “in a better place”.
Against Tsitsipas, it certainly was. And at a level where it has rarely been – if at all – since Murray first developed the hip problem at the 2017 French Open.
Playing with freedom of movement and plenty of power, Murray’s quality drew numerous errors from Tsitsipas as he dominated the opening set and threatened to take a two-set lead. (Courtesy BBC)