Tsonga, 34, who made the Wimbledon semi-finals in 2011 and 2012, is now ranked at 72 in the world after undergoing knee surgery last year.
French Open champion Nadal is chasing a 19th title at the majors having recently captured a 12th Roland Garros crown.
Victory on Saturday was achieved in just 1 hour and 48 minutes and was a much less raucous affair than his bad-tempered second round duel with Nick Kyrgios.
Men's singles third round:
- Sam Querrey (USA) - John Millman (AUS) 7-6 (7/3), 7-6 (10/8), 6-3
- Tennys Sandgren (USA) bat Fabio Fognini (ITA, 12th seed) 6-3, 7-6 (14/12), 6-3
- Kei Nishikori (JPN, 8th seed) - Steve Johnson (USA) 6-4, 6-3, 6-2
Women's singles third round:
- Ashleigh Barty (AUS, 1st seed) - Harriet Dart (GBR) 6-1, 6-1
- Alison Riske (USA) - Belinda Bencic (SWI) 4-6, 6-4, 6-4
- Serena Williams (USA) - Julia Görges (GER) 6-3, 6-4
- Carla Suárez (SP) - Lauren Davis (USA) 6-3, 6-3
- Barbora Strýcová (CZE) - Kiki Bertens (NED, 4th seed) 7-5, 6-1
- Elise Mertens (BEL) - Qiang Wang (CH) 6-2, 6-7 (9/11), 6-4
- Petra Kvitová (CZE, 6th seed) - Magda Linette (POL) 6-3, 6-2