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Former world number one singles players Serena Williams and Caroline Wozniacki also breezed into the semi-finals of the doubles after they thumped top seeds Caroline Dolehide and Johanna Larsson 6-2 6-1 in just 52 minutes.
Wozniacki and Williams, who are good friends off the court, have teamed up to play as a doubles pairing for the first time before the Dane retires after the Australian Open.
"Multiple times I was thinking 'wow'," Wozniacki said of the power strokes from Williams that blew the American/Swedish duo off court.
"I just wanted to win today. I'm happy for my second doubles victory in 10 years," the Dane added while laughing.
Both will play their second-round singles matches on Thursday.
Bouchard, a former top-five player who made the Wimbledon final and semi-finals at the Australian Open and Roland Garros in 2014, has slumped to 262 in the rankings and is attempting to resurrect her career.
The 25-year-old Canadian battled for every point in her 6-4 6-4 second round victory against the Frenchwoman, who is mired in her own form slump having failed to advance past the second round of her previous 12 tournaments.
Bouchard will now face either promising American teenager Amanda Anisimova or Russia's Daria Kasatkina in the quarter-finals.
France's Cornet also provided something of an upset when she beat Croatian second seed Martic 3-6 6-4 6-4 in their second round clash and sealed the win with her only three aces of the match in the final game after she had gone 0-30 down.
"I lost the first set and it was not easy, but I came out of my comfort zone and I did a lot of stuff that I'm proud of," Cornet told reporters.
"I got through and that's the most important thing."