Berdych defeated Lu Yen-Hsun 7-6 (9-7), 6-4, winning only six more points overall than Lu but getting two breaks to one for Lu. He won two-thirds of the points on serve, a statistic bolstered by 20 aces in 11 serving games.
Gasquet, the defending champion, got by fifth-seeded Mikhail Youzhny 5-7, 6-3, 6-1. Gasquet dropped his serve three times overall but lost just two points when serving in the third set.
Raonic needed only 58 minutes to get by sixth-seeded Feliciano Lopez 6-4, 6-3. Raonic had 19 aces and dropped eight points in nine service games.
Simon edged past Igor Sijsling 6-4, 5-7, 6-4 in 2 1/2 hours. He saved 14 of Sijsling's 16 break opportunities, including two in the third set.
That put Simon against Berdych in Saturday's first semifinal. Simon is 5-2 versus Berdych but they've only played once since 2009.
Gasquet and Raonic have split their two career meetings. Gasquet took a five-set thriller in the fourth-round of this year's U.S. Open in the most recent match.