Edmund, ranked 84th, beat former semi-finalist Gasquet 6-2 6-2 6-3 to claim the best win of his career on a hot afternoon at Flushing Meadows.
The Briton, 21, will face American wild card Ernesto Escobedo in round two.
British number one Konta, seeded 13th, was in superb form as she beat Mattek-Sands 6-3 6-3 in the evening session.
The 25-year-old goes on to face Bulgaria's Tsvetana Pironkova in round two.
Andy Murray is among six Britons in singles action on day two, with the Scot taking on Czech Lukas Rosol at around 02:00 on Wednesday.
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.
"It's definitely one of my best wins in my career," Edmund said, who added that working with Andy Murray in the Davis Cup set-up had helped improve his overall game.
"I had a good summer. I beat some good players, I beat [Gilles] Simon and then had a good experience against Andy [at Queen's].
"Those matches against top players give you more experience, more confidence with your game, that you are able to take it to them."
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.
"It's definitely one of my best wins in my career," Edmund said, who added that working with Andy Murray in the Davis Cup set-up had helped improve his overall game.
"I had a good summer. I beat some good players, I beat [Gilles] Simon and then had a good experience against Andy [at Queen's].
"Those matches against top players give you more experience, more confidence with your game, that you are able to take it to them."
