Not a fan of the new Sunday start, Thompson looked on track for a swift return to Manhattan before being frustratingly waylaid in a 6-2 6-4 1-6 6-3 victory over plucky Frenchman Corentin Moutet.
But after dropping the third set and turfing his racquet after falling break point down midway through the third, Thompson regrouped to bring up match point with a stylish, Pat Rafter-like backhand smash.
He also closed out the contest at the net to book a second-round date with another French leftie, Adrian Mannarino.
Earlier, Thompson's fellow Sydneysider Rinky Hijikata made a hasty exit with a 6-2 6-1 6-2 loss to Italian 32nd seed Luciano Darderi.
An underdone Hijikata preferred to draw the positives from an otherwise "horrific" first-round defeat.
While Darderi has been riding a hot streak since Wimbledon, collecting back-to-back clay-court titles in Bastad in Sweden and Umag in Croatia, Hijikata has been recovering from a torn abdominal muscle.
The lay-off was evident as Hijikata lost his comeback match at Flushing Meadows in 85 minutes.
"Pretty horrific today," he said.
"Luciano played well. I didn't really do anything to disrupt or make him uncomfortable.
"It's tough coming straight back in first match pretty much since 'Wimby'.
"Just starting here is not easy. Obviously against a quality opponent too, it's not like you ease your way into the match against him. He's hitting the ball massive from the back and serving big."
Hijikata sustained his injury during his run to the Wimbledon doubles final six weeks ago with Dutchman David Pel.
"It was a bit of a Catch 22," Hijikata said.
"It was unbelievable two weeks and I probably wouldn't trade it. A final of Wimbledon 'dubs', It's not something you can do all the time, and who knows if I make that stage again.
"So I don't regret playing. It was great, a lot of fun playing.
"But, yeah, it definitely made it tricky just because I thought I was going to have a bit more of a gap after my singles at Wimbledon to try to make sure I was 100 per cent and get ready for the US hardcourt swing."
Hijikata made the last 16 in New York two years ago but will depart this year with a provisional live ranking of world No.107 and plenty of work to do to gain direct entry for the Australian Open in January.
"The US swings have historically had good success," the Sydneysider said.
"I was excited to play in this swing, but I guess it's tricky. That comes with the risk of playing both singles and 'dubs'.
"To finish the year I definitely want to make sure that my body's healthy, that I'm prioritising my singles, and I've got a lot of work to do now."
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