The Greek, runner-up in Paris in 2021, did not expect the kind of resistance offered by a player ranked 445th in the world following a lengthy injury absence, and had to save four set points in the fourth to avoid a decider.
"I said to myself there's no chance this is going to a fifth set. That's what I told myself," Tsitsipas said. "Jiri was a difficult obstacle. He gave me a hard time and I am happy I overcame it in such a fashion.
"Today's win is a very important for me," said Tsitsipas, who has not enjoyed his best clay court season. "I felt at times I was not going to the ball, I was staying still. When I took charge that was the moment I made that switch and won the match."
Tsitsipas was broken in his second service game and found himself 5-3 down against Vesely, playing in his first tour-level tournament since last year's U.S. Open.
But he broke the 29-year-old twice in a row to clinch the next four games and close out the first set.
Tsitsipas had initially struggled with the tall left-hander's awkward spin but was now stretching his opponent with deep cross-court forehands, bagging the second set with another break.
The world number five, chasing his Grand Slam title, thought he had hit his stride, comfortably holding serve in the third but Vesely, who has played in only two Challengers since returning last month, doggedly refused to budge.
He snatched the third set on his first opportunity with frustrated Tsitsipas sinking an easy forehand into the net on set point before the pair traded early breaks in the fourth.
There were no signs of rustiness for Vesely and the Czech kept up the pressure to earn four set points in the tiebreak.
He could not convert them, however, allowing Tsitsipas to clinch victory on his first match point with another sizzling crosscourt forehand winner after three hours and 13 minutes.
