Wawrinka, seeded seventh in Paris, defeated 12th-seeded Nicolas Almagro 6-2, 6-2 and later Gasquet, seeded ninth, topped Kei Nishikori 6-3, 6-2 and advanced to the BNP Paribas Masters quarterfinals.
The wins were important but, in fact, the invitations to the Tour Finals in London came when sixth-seeded Tomas Berdych beat Milos Raonic, the 10th seed, 7-6 (15-13), 6-4. By Thursday Raonic was the only player, other than those who ended up qualifying, with a chance at a berth in London, but he needed to reach the finals in Paris to do so.
The rest of the Paris Masters will be something of a London preview as the eight quarterfinalists make up the field in London.
World No. 1 Rafael Nadal will play Gasquet and third-seeded David Ferrer takes on Berdych in the top half of the bracket's quarters Friday while No. 2-seeded Novak Djokovic plays Wawrinka and fourth-seeded Juan Martin del Potro meets No. 5-seed Roger Federer in the other.
Nadal wrapped up third-round play with a 7-5, 6-4 victory over 14th-seeded Jerzy Janowicz. Nadal had the break late in the first set to take the lead and had a pair in the second set, with the second ending on a double fault by Janowicz.
Djokovic started relatively slowly but still beat 13th-seeded John Isner 6-7 (5-7), 6-1, 6-2 without facing a break point, and Ferrer, who also didn't lose his serve Thursday, ousted No. 15 Gilles Simon 6-2, 6-3.
Del Potro was a 3-6, 6-3, 6-4 winner over Grigor Dimitrov, and Federer improved to 8-0 all-time against Philipp Kohlschreiber with a 6-3, 6-4 victory.
Swim Week Miami Beach 2014 [PHOTOS]