The Frenchman, who has won the event twice before when it was held in Nottingham, will meet Denis Istomin of Uzbekistan in the last four.
"I've won two good matches," said Gasquet who celebrated his 28th birthday on Wednesday. "I served well and could play with power."
Istomin overcame Gasquet's experienced 30-year-old compatriot Edouard Roger-Vasselin 7-6 6-2.
Big-hitting American Sam Querrey defeated a third Frenchman, Julien Benneteau, 7-6 (5) 6-4 and next meets the winner of the duel between defending champion and Queen's Club runner-up Feliciano Lopez and another Frenchman Jeremy Chardy.
Their match was postponed when a fine drizzle started to fall on the English coastal resort.
In the women's competition, fifth seed Angelique Kerber of Germany strode into the semi-finals with a 6-2 6-1 win over Ekaterina Makarova.
"It was a really tough match," said eighth seed Wozniacki. "She hits the ball very hard and flat and has some good serves."
The Dane will face Kerber in the semi-finals and they will be joined by American Madison Keys who beat compatriot Lauren Davis 6-2 6-1.
Keys will be up against home favourite Heather Watson who had a walkover against second seed Petra Kvitova after the Czech withdrew with a leg injury.
Kvitova, the 2011 Wimbledon champion, said she hoped to be fit for the grand slam starting on Monday.
- Sports & Recreation
- Denis Istomin
- Angelique Kerber
- Julien Benneteau