The two-time Wimbledon champion, a winner here in 2012, overcame an unorthodox and awkward opponent 6-2 7-5 6-3 to become the eighth male player in the Open era to reach the double-century.
It may have been a straight-sets win but, typically with Murray, there was plenty of drama and even a VAR controversy, the first of its kind at Flushing Meadows.
Murray, who came into the event having recovered from an abdominal tear, dominated the first set but a flat passage of play saw him fall a break down in the second as Moutet, all slices and drop shots, began to impose himself.
However, Murray had a let-off when Moutet double-faulted on set point, and then a lucky net cord helped the Scot break back.
Murray was back in full irritant mode and it got to Moutet, who smashed his racket on the floor, twice, as the second set got away from him.
Moutet suffered a nasty fall on the baseline, landing on his racket hand, but was able to continue and saved four break points in his next service game.
But Murray struck at the next opportunity and served out for victory - following a video replay check for a double-bounce which, embarrassingly for organisers who introduced it this year, proved inconclusive - in just under three hours.
"He's one of the most skilful players on the tour, with so many ways to disrupt you, and he always causes a little bit of chaos," said the 36-year-old.
"I hope it was entertaining, there were some fun points, so I'm happy to get through in straight sets.
"It was a long one, but the way we play it was probably always going to be like that. Three hours is a lot shorter than some of my matches!"
Murray will meet Bulgarian 19th seed Grigor Dimitrov, who saved three match points in a five-setter against Alex Molcan, in the second round.