Gasquet, who has reached the last 16 at Flushing Meadows four times in his career, was outplayed throughout and his US Open experience lasted just one hour, 41 minutes.
The 21-year-old Edmund, who helped Britain win the Davis Cup last year, crunched 10 aces and 40 winners in all en route to the biggest victory of his career.
Edmund will play Lukas Lacko of Slovakia or American wildcard Ernesto Escobedo in the second round.