The top three seeds were among the players handed first-round byes at the Masters 1000 event, and Djokovic and Murray were quick to adapt to the hard courts in Canada after their Olympics exploits on the Wimbledon grass.
Serbian top seed Djokovic, a losing semi-finalist in SW19, breezed past Bernard Tomic, breaking the Australian teenager's serve twice in the first set and once in the second en route to a 6-2 6-3 victory in one hour 11 minutes.
Olympic champion Murray, seeded two, enjoyed an equally straightforward passage into round three, toppling Italian Flavio Cipolla 6-1 6-3 in one hour 21 minutes to set up a clash with home favourite Milos Raonic.
It was a different story for Tsonga, though, as Chardy overcame the world number six 6-4 7-6 (7/4).
Sixth seed Juan Martin Del Potro, the bronze medallist in London, was another surprise casualty, Czech Republic's Radek Stepanek sinking the Argentinian 6-4 7-6 (7/5).
Stepanek's compatriot Tomas Berdych, seeded four, also advanced but did it the hard way against Frenchman Julien Benneteau - 6-7 (10/12) 6-4 6-4.
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Fifth seed Janko Tipsarevic, 10th seed Marin Cilic, 12th seed Philipp Kohlschreiber and 14th seed Richard Gasquet also progressed thanks to straight-sets triumphs over Mikhail Youzhny, Marcos Baghdatis, Fabio Fognini and Mikhail Kukushkin, respectively.
Several other seeds also fell by the wayside on Wednesday, though. Germany's Tommy Haas saw off ninth seed Gilles Simon of France 6-2 6-3, Sam Querrey defeated 13th-seeded Japanese Kei Nishikori 6-2 6-3 - the American will face Djokovic next - and Spain's Marcel Granollers overcame 15th seed Florian Mayer of Germany 6-3 6-4.