|US Open 2019|
|Venue: Flushing Meadows, New York Dates: 26 Aug – 8 Sep|
|Coverage: Live text and radio commentary on selected matches on the BBC Sport website and app|
The 23-year-old, who has reached three consecutive tour finals, took just 85 minutes to win.
Medvedev will face South Korean Kwon Soon-woo or Bolivian Hugo Dellien in the second round – both ranked outside the world’s top 80.
Gunneswaran, ranked 88th, was playing in the main draw for the first time.
Elsewhere, Japanese seventh seed Kei Nishikori raced into the second round when he took a 6-1 4-1 lead before Argentine opponent Marco Trungelliti, 29, retired with a back problem.
“It’s a little bit sad to see. He’s a great player,” said Nishikori, who faces 108th-ranked American Bradley Klahn in the next round.
“I played great Tennis from the beginning and I’m happy with the way I played. But honestly, I wanted to play a little more because I was feeling good on the court and I wanted to get a little more confidence.”