Rublev was serving for the match at 5-4, 40-30 but hit a forehand narrowly wide. Fifth-seeded Thiem broke him with backhand pass down the line and held for 6-5.
The Austrian was 15-40 up on Rublev’s serve and clinched victory on his first match point, when Rublev double-faulted with a weak serve into the net.
“I was 10 centimeters from being out of the tournament,” a relieved Thiem said. “But I’m happy that I played two hours and 40 (minutes).”
Thiem has reached the French Open semifinals for the past two years. He next meets 12-time Grand Slam champion Novak Djokovic or Borna Coric of Croatia, who play their second-round match on Wednesday.
“I’m looking forward to watching Djokovic and Coric in front of the TV, and then playing the winner on Thursday,” Thiem said.
In the second round later Tuesday, fourth-seeded Grigor Dimitrov of Bulgaria faced Pierre-Hugues Herbert and seventh-seeded Lucas Pouille played Mischa Zverev.
In remaining first-round play, there were wins for Gilles Simon of France, Marco Cecchinato of Italy and Jan-Lennard Struff of Germany.