The British number one dispatched his Australian opponent 6-3 6-4 in a rain-delayed match.
A passing shower halted the first set at 4-4, with Edmund returning to comfortably dominate Kyrgios and move into the next round.
— ATP Tour (@ATP_Tour) August 7, 2019
Edmund told the ATP's official website: "It was obviously very tricky playing Nick... He's got a very good serve, probably one of the best on Tour. It's a mini battle in itself, trying to get that back.
"I managed my game well, I served very well today. I thought I just controlled the points with balls in play. It was a really good win for me."
Edmund will next meet the other Washington finalist, Daniil Medvedev.
Dan Evans will meet Rafael Nadal for the first time after easing his way into the second round with a straight-sets victory over Australia's Alex de Minaur.
Evans took the first set 6-4 but was pushed further in the second before closing it out 7-6 (6), having saved seven of eight break points he faced in the match and converting both of those which came his way.
Cameron Norrie, though, was beaten 7-6 (4) 6-4 by third seed and 2017 champion Alexander Zverev.
The British number three played a strong part in the first set in particular but was eventually beaten in a match featuring some entertaining rallies.
Andy Murray and Feliciano Lopez held on to save match point and win a tight doubles opener against Lukasz Kubot and Marcelo Melo.
The Scot and his Spanish partner were a wildcard entry but managed to down the second seeds 6-7 (5) 6-3 11-9.
The pair will next face France's Jeremy Chardy and Fabrice Martin.
Jamie Murray and Neal Skupski made a swift exit as they were beaten 7-5 7-5 by fellow Briton Joe Salisbury and his American partner Rajeev Ram.