Edmund, ranked 84th, beat the former semi-finalist 6-2 6-2 6-3 on a hot afternoon at Flushing Meadows.
The Briton, 21, will face American wild card Ernesto Escobedo in round two.
British number one Johanna Konta plays American Bethanie Mattek-Sands later on Monday, with Andy Murray among six Britons in singles action on Tuesday.
I played better than I expected – Edmund
Edmund bullied former semi-finalist Gasquet with his huge forehand on his way to a two-set lead, as the Frenchman struggled in 32C heat on court five.
There was a glimmer of hope for Gasquet when he broke serve first in the third set but a brilliant running pass sparked a five-game run that took Edmund over the finish line.
“I played really well, went about my process and didn’t really worry about who I was playing,” Edmund told Eurosport. “I probably played better than I expected today.
“I always know in best-of-five sets there’s up and downs and I was obviously riding a high until I got broken in the third set.
“When I broke him at 3-2, I thought I’d really push it home and take it to him.”