The 2021 champion fired off 14 aces in a lopsided affair, where a mid-match avian distraction served up an odd bit of drama.
Medvedev next plays either Spain's Jaume Munar or Frenchman Arthur Rinderknech.
"Being on the court, I was happy with my level," the Russian said after the match. "I love this court and I love you guys."
The Russian broke his opponent from the baseline in the opening game and needed only 21 minutes to get through the first set, firing off 10 winners to just two from Gorzny, who was outmatched in only his second major main-draw match.
Medvedev has kept a calm demeanour in New York - in contrast to his dramatic first-round exit last year - but had his patience tested when a pair of birds appeared on the net during a rally midway through the second set, prompting the umpire to replay the point.
"I saw them straight away and thought the point would stop straight away," said Medvedev, who estimated they continued to play for 45 seconds before the umpire intervened.
"The moment I hit the winner we stop and I say why?"
He got over it pretty quickly, keeping the demolition job going in the third set, where he briefly sniped at the chair umpire, calling him "crazy," during the changeover after the third game and survived a 33-shot rally in the sixth.
Gorzny improved his level in the third set, saving four break points in the ninth game, but it was too little too late as Medvedev harnessed his powerful forehand to break the American in the 11th game and sealed it with an unreturnable serve.
Later, Rafael Jodar was tripped up at the first hurdle of his maiden US Open campaign as the Spanish 12th seed failed to find his rhythm and fell 6-2 6-1 6-1 to Chinese lucky loser Bu Yunchaokete.
Back at the site of his junior triumph two years ago, the 19-year-old Jodar looked unusually out of sorts after his first-round clash was delayed by a day due to rain, with a flurry of unforced errors handing Bu the opening set.
Things looked to be going from bad to worse for Jodar when Bu, the last-minute beneficiary of a spot in the main draw after Australian Thanasi Kokkinakis withdrew, broke early in the second set and threatened to run away with the contest.
"I'm a really lucky loser," Bu said.
"After I lost in qualifying, I took a day off but I saw a few guys pull out, so I took the decision to come back and tried to do some work in case I could play. Yesterday, they told me I could ... when I heard that, I was super excited."
Ben Shelton survived a tough test from Poland's Hubert Hurkacz 6-3 5-7 7-6 (7-3) 7-5 before reaching the US Open third round, relying on the deafening cheers of the partisan American crowd to power him through.
The eighth seed, bidding to end a 23-year US men's major title drought, arrived in New York on the back of his Montreal victory and enjoyed strong backing in Arthur Ashe Stadium during the finely poised encounter.
