After the second-seeded Tsonga beat Nikolay Davydenko 7-6 (4), 7-6 (8), his French countryman Gasquet followed up with a 6-3, 6-3 win against Slovenia's Blaz Kavcic. Gasquet did not face a single break-point and took his opponent's serve three times.
Earlier, the second-seeded Tsonga, who had 19 aces, improved to 4-2 in career head-to-heads against the Russian who reached a career-high ranking of third.
Davydenko saved one match point with a forehand pass down the line, but Tsonga clinched victory with the next when Davydenko's forehand hit the net.
"It's never easy against him, he's a great player who's won a lot of titles," the 28-year-old Tsonga said. "I'm trying to get my bearings again, which is why I was so happy to win."
Tsonga next plays either sixth-seeded Edouard Roger-Vasselin of France or Ukraine's Sergiy Stakhovsky, while Gasquet's quarterfinal opponent is either fifth-seeded Ivan Dodig of Croatia or Germany's Tobias Kamke.
"I didn't play well last time (in Rotterdam). But you shouldn't forget that, and use the experience," said Tsonga, referring to his straight-sets loss to Croat Marin Cilic in the second round of the ABN Amro tournament. "I absolutely must raise my level."
In the remaining first-round matches on Wednesday, there were wins for seventh-seeded Julien Benneteau of France, No. 8 Nicolas Mahut of France, Spain's Roberto Bautista Agut, Germany's Jan-Lennard Struff, Frenchman Michael Llodra and Ricardas Berankis of Lithuania.
Benneteau of France won 6-1, 6-4 against Jesse Huta Galung; Mahut beat Kenny De Schepper 7-6 (2), 6-3; Struff beat David Guez 3-6, 6-1, 6-1; Bautista Agut won 7-5, 4-6, 6-3 against Dustin Brown; Llodra downed Lukas Lacko 6-3, 7-6 (2), and Berankis led 6-5 when Jan Hajek retired.