The German was in sublime form, needing just 70 minutes to dispatch the wildcard 6-2, 6-3 to get his campaign underway. Zverev crushed 16 winners and was completely dominant at the net.
He had few problems behind his big serve, never getting to break point, while Damm scrapped to save five out of the break points he faced, but it was not enough.
"He's somebody that's extremely dangerous, has a massive serve, massive game," Zverev said of Damm after their first-ever meeting.
"I'm happy with the match, happy with my level, and looking forward to what's ahead."
Elsewhere, Alexander Shevchenko sent eighth seed Ben Shelton to another early exit with a superlative victory. The Kazakh came back from set down to win 6-7(3), 7-6(3), 6-3 for his third-ever victory over a top ten player
"It takes 100 per cent to stay in the match all the time against these kind of players," said Shevchenko afterwards.
"There was a moment that I was feeling a little bit tired, but as soon as I got the momentum back, I was bringing the energy back."