The 2012 US Open victor once again showed his mettle, putting pressure on Gasquet's serve and holding his next two to make it 4-3 with his game improving all the time.
Andy Murray moved well in his first singles match since January, but not well enough to move on - even to the U.S. Open.
The Scot added that he might potentially play singles next week in Winston-Salem, but that he plans to play men's doubles and mixed doubles in Flushing Meadows.
"We were hoping to maybe hold a wild card until a little bit closer to the time to see how I feel and get some matches hopefully and a bit of practice", the 32-year-old told reporters.
However, he had been asked to make a decision on accepting a wildcard for the US Open prior to facing Gasquet.
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Gasquet will face 4 seed Dominic Thiem of Austria in the second round. "I think I did OK". "That's just me not running for a ball, which I did do that better at the end of the match".
It was his first singles match in seven months.
"I just didn't know today how I was going to feel after a match", said Murray, who indicated US Open organizers wanted to know whether he was interested in a singles wild card before he played Gasquet.
American John Isner, the 13th seed, also progressed, defeating Serbia's Dušan Lajović 7-6, 1-6, 7-5. "I'm trying to create more things like that".
In the women's draw, five-time Grand Slam victor Maria Sharapova moved on to a second-round matchup with top-seeded Ashleigh Barty by knocking off fellow wild card Alison Riske 6-3, 7-6 (4).
This year, Prime Video will broadcast on-site, live studio coverage across tournaments, including at the BNP Paribas Open from Indian Wells, the Fever Tree Championships from The Queens Club, the US Open, and the Nitto ATP Finals from The O2.
In another upset in this U.S. Open tuneup, 13th-seeded Angelique Kerber was knocked out by unseeded Anett Kontaveit 7-6 (7), 6-2, and 15th-seeded Qiang Wang lost to Su-Wei Hsieh 6-3, 6-4.