Gasquet, seeded No. 1, topped Blaz Kavcic 6-3, 6-3, losing just 11 points over nine games serving. Gasquet didn't see a break point and won three of the 11 break chances he forced on Kavcic.
Tsonga is seeded second as he tries to defend his 2013 Open 13 title but he had get through a tight 7-6 (7-4), 7-6 (10-8) battle with Nikolay Davydenko. Tsonga won five more points (85-80) overall but those extra points all came in the tiebreakers.
Both players won more than 70 percent of the points on serve and dropped serve just once.
Tsonga also beat Davydenko in the second round last year at Marseille.
The first round concluded with seventh-seeded Julien Benneteau beating Jesse Huta Galung 6-1, 6-4 by winning three-quarters of the points on serve and No. 8-seeded Nicolas Mahut stopping Kenny De Schepper 7-6 (7-2), 6-3 without facing a break point.
Also Wednesday, Roberto Bautista Agut got by Dustin Brown 7-5, 4-6, 6-3; Jan-Lennard Struff rallied for a 3-6, 6-1, 6-1 decision over David Guez; and Michael Llodra advanced over Lukas Lacko 6-3, 7-6 (7-2). Ricardas Berankis took a spot in the second round when, with Berankis ahead 6-5 in the first set, Jan Hajek retired.