The 31st-ranked Frenchman, a former world No. 7, battled back from a break down to level at 4-4 in the second set but dropped his serve again at 5-5. Pospisil served out the match without conceding a point.
Earlier, fifth-seeded Stepanek outlasted Lukas Lacko of Slovakia 7-6 (4), 5-7, 7-6 (5) to reach his second quarterfinal of the season after losing in the last eight at Eastbourne in June.
"It's all about practice and preparation. This is the result," Stepanek said after the match lasted for more than 2 hours. "It was very close."
Stepanek saved two set points while serving at 5-4 down in the opening set. After leveling the match, Lacko received treatment for a right thigh injury. Stepanek missed his first match point at 5-4 in the final set before closing out the victory with his 15th ace.
"In the second set, he got physical troubles but I didn't know how to handle the situation," said Stepanek with a hoarse voice. "You can hear that I am not 100 percent healthy either."
Eighth-seeded Rosol served 10 aces while beating Mirza Basic of Bosnia-Hercegovina 6-3, 7-5 to reach his third quarterfinal of the year.
In first-round play, Lukasz Kubot of Poland defeated Florian Mayer of Germany 7-6 (1), 6-4. The 74th-ranked Kubot, who came back from 3-0 down in the second set, next plays third-seeded Fabio Fognini of Italy.