Edmund never faced a break point against Kyrgios, who won the Citi Open in Washington on Sunday despite an ailing back, which seemed to limit his movement in the match.
Kyrgios and the chair umpire got into an extended argument about the availability of court-side towels early in the match but Edmund kept his focus to put on a serving masterclass.
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Next up for Edmund is a second-round meeting with rising eighth seeded Russian Daniil Medvedev on Wednesday.
British qualifier Dan Evans faces a tough second-round clash against top seed and defending champion Rafa Nadal after beating Australian Alex de Minaur 6-4 7-6(6).
Big-serving American John Isner fired down 26 aces to help secure a 6-3 3-6 7-6(6) win over Australian Jordan Thompson.
The 12th seed, who had lost three of his four tiebreaks in the two weeks leading into the event, put an end to the tense battle when he fired down an ace on his third match point to set up a clash with Chilean Cristian Garin.
Canadian teenager Felix Auger-Aliassime also secured a spot in the second round of the U.S. Open tune-up with a 6-2 6-7(3) 7-6(3) win over compatriot Vasek Pospisil.
Awaiting 18-year-old Auger-Aliassime is 17th-seeded fellow Canadian Milos Raonic, the latter having won their only career meeting last year in Indian Wells.
Third seeded German Alexander Zverev won 80 percent of his first serve points to dispatch Cameron Norrie 7-6(4) 6-4 and set up a second-round meeting with Nikoloz Basilashvili, who took out Jan-Lennard Struff to advance.
Croatia's Marin Cilic, the 14th seed, got past American qualifier Bradley Klahn 6-3 7-6(7) to advance, while Argentine Guido Pella upset Belgian 15th seed David Goffin 6-4 7-6(4).
In other first-round matches, Moldova's Radu Albot beat Frenchman Gilles Simon while Argentina's Diego Schwartzman and Australian lucky loser John Millman also advanced.
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