"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.
But former world number one Caroline Wozniacki struggled to overcome a determined Camila Giorgi of Italy 6-7 6-4 6-2 in two hours 35 minutes.
"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.
(Reporting by Clare Lovell, editing by Tony Goodson)
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