Haase lost the opening set as Verdasco converted the only break point, but he broke the Spaniard at 5-4 in the second set. Haase lost his opening service game in the third before winning six of the following eight games to wrap up the win.
"It was very tough and very hot," Haase said. "He was the better player in the first set. I didn't have a chance. From the end of the second set, I managed to put him under some pressure."
After winning in 2011 and `12, Haase is now 12-0 at the event. He beat Granollers on a third-set tiebreaker in the quarterfinals in Gstaad a week ago.
Granollers had five aces and held serve throughout to beat Leonardo Mayer of Argentina 6-3, 6-1 and reach his first semifinal of the season.
The second-seeded Monaco advanced to his third semifinal of the year by defeating Daniel Gimeno-Traver of Spain 6-2, 3-6, 6-4. The Spaniard had beaten Monaco from two sets down at Roland Garros in May, but couldn't complete a comeback this time.
"It wasn't great, but I am just happy to go through," said Monaco, who won the event in his last appearance in 2007. "From now on, I have nothing to lose."
Montanes defeated Dominic Thiem 6-4, 6-3 in the 19-year-old Austrian's first career quarterfinal.