The 98-minute first-round win sends Tsitsipas into a match with Brazilian Joao Fonseca.
The 53rd-ranked Tsitsipas, winner last month in Gstaad, said that a coaching change -- he split from his father Apostolos for a second time - is paying off now that he is under the guidance of coaches from Patrick Mouratoglou's French academy.
"It's a chapter that should have started a long time ago," Tsitsipas said. "I guess I was too patient with my time and too nice with handling certain things, to take so much time to proceed with the change. But it's how life works."
"Certain things come at a certain time, I'm very Buddhist in that sort of philosophy.
"Now we are here, and we are doing it the way I like it. I want it to be done. I want it to be done with a coach that I am happy with right now ... someone that I trust."
The Greek claimed the opening set against Damm and came back from a break down in the second to earn a 5-2 lead and the chance to serve it out.
But the Czech managed to break back, forcing Tsitsipas to clinch victory two games later on a forehand winner.
"I need to focus and concentrate, and that's what I was trying to do out there today, trying to keep it locked in as much as I could and not let any distractions get in the way," Tsitsipas, 27, said.
"Tennis is about milliseconds, getting properly behind the ball and being constantly well positioned. I need to focus on how to do that in the best possible way.
"I felt in control up two breaks, but things got tough. I have nothing to regret, I was able to fight my way through the match. I took a good grip to close it out."
Hubert Hurkacz a 2022 finalist in Montreal who is coming back from a knee injury suffered at Wimbledon, returned with a 7-5, 4-6, 6-2 win over Marcos Giron.
Britain's Jakob Fearnley, who reached the final of a hard court Challenger last month, advanced past French veteran Adrian Mannarino 7-6 (7/3), 6-4, overcoming more than 30 unforced errors to book a meeting with 13th seeded Czech Jakub Mensik.