Pouille beat eighth-seeded Filip Krajinovic of Serbia 7-6 (5), 3-6, 6-4, hitting 20 aces and securing victory on his first match point when he broke Krajinovic's serve for the first time.
Berdych led 7-6 (4) when seventh-seeded Damir Dzumhur of Bosnia-Herzegovina pulled out feeling sickly.
"He was ill all night, it can't be easy to play in the shape he was in," Berdych said. "I was in the same situation last week."
Pouille is looking for his second straight title after winning the Open Sud de France two weeks ago and plays 193rd-ranked qualifier Ilya Ivashka of Belarus in the semifinals. Berdych will face Karen Khachanov of Russia, who beat Berdych in the second round of the French Open last year.
In their quarterfinals, Khachanov had 16 aces in beating local Julien Benneteau 6-4, 6-4 and Ivashka beat another Frenchman, Nicolas Mahut 4-6, 6-3, 7-5. Mahut saved 10 of 13 break points against Ivashka, who had 13 aces.