The top seed needed just 69 minutes to secure a 6-3 6-4 triumph over Klizan at ATP 250 grass-court event in Eastbourne.
Klizan has risen from world number 108 to 57 since the start of the season, but was unable to get the better of the Frenchman.
"I have won two good matches," said Gasquet. "(Bernard) Tomic and Klizan are good players. I served well (and) I could play with power. It's a good start of the week for me."
Istomin's route to the last four was lightly trickier, but the Uzbek came through 7-6 (7-5) 6-2 against France's Edouard Roger-Vasselin.
American Sam Querrey proved his credentials on grass by beating compatriot Mardy Fish in the final of 2010 Aegon Championships.
And he edged closer to another title on the surface by defeating Julien Benneteau 7-6 (7-5) 6-4.
The 26-year-old was boosted by 16 aces en route to victory in a time of one hour and 22 minutes.
Third seed Feliciano Lopez's match with Jeremy Chardy, meanwhile, was postponed due to rain.