The win also represents a huge milestone for Murray as it was his 200th singles victory at a Grand Slam. He is now three wins behind Pete Sampras on the all-time leaderboard.
Murray will play world No.19 Grigor Dimitrov in the second round.
‘It was amazing, a brilliant atmosphere. He [Moutet] is one of the most skilful players on the tour. He has so many ways to disrupt you,’ Murray said after his win in New York.
‘He always causes a little bit of chaos. I hope it was entertaining for everyone.
‘I am glad to get through in straight sets because the second set was very tight. It was a long one but the way that we play it was probably always going to be that.
‘It was a lot shorter than most of matches this year! I’m happy to get it done in under three hours.
‘I had to slowly build my serving up this month but I was happy with how I served today. I can’t wait to move on to the next one.
‘I’m at my highest ranking since I had the metal hip put in. I’m happy with that. It’s not been an easy journey. Obviously I want to be ranked higher but this is the best I’ve played consistently since 2017.
‘I’m happy to keep going and progressing and am hoping to push my ranking up.’
Elsewhere, the current British men’s No.1 Cameron Norrie beat Alexander Shevchenko 6-3 6-2 6-2 in his first-round match.
He will play Taiwanese qualifier Yu Hsiou Hsu, ranked 247 in the world, in the second round.
Meanwhile, Katie Boulter reached the US Open second round for the first time in her career as she won 6-4 6-0 against her 20-year-old French opponent Diane Parry.
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