Australian Open runner-up Tsitsipas lost the third set and had to save four set points in the fourth set tie-break to seal a 7-5 6-3 4-6 7-6 (9-7) victory over his 455th-ranked Czech opponent.
He will play Spain's Roberto Carballes Baena or American Emilio Nava next.
"I said to myself there's no chance this is going to a fifth set," Tsitsipas, 24, said.
"Jiri was a difficult obstacle. He gave me a hard time and I am happy I overcame it in such fashion."
Vesely only returned to competition in April after injuring his hamstring during last year's US Open.
The 29-year-old, who has contested just two ATP Challenger events since then, threatened to force a fifth set against Tsitsipas but failed to take any of his opportunities in the tie-break, allowing the Greek to rally from 6-3 down.
Tsitsipas, who lost to Novak Djokovic in January's Australian Open final, has not lost in the first round at Roland Garros since 2017.