Cornet's last win over a player ranked in the top five in the world was also a hard court win over Kerber in Beijing in 2017.
The third-seeded Stephens had little trouble dispatching the last Canadian left in the singles draw with a 6-0, 6-2 victory on Wednesday.
Azarenka punched her ticket to the second round with a dominating 6-0, 6-1 victory in just 55 minutes over the former world number 10.
"I came on court having a lot of respect for my opponent because she's up and coming but has already established herself", said the 31-year-old Sharapova. "I would next time try to impose my game more, try to be more aggressive".
"I came in with a similar mindset [to yesterday], just a focused mindset that I needed to carry on to today's match", Sharapova told the press afterwards.
The Russian had a good start to her tournament despite having to play it in two parts.
An unbelievable drop shot victor set up match point for the world number 34, but Kerber was determined to put up a tough fight as she sent in a backhand down-the-line victor to stay alive. "I'll take what I can get as long as I'm the one getting a chance to play the next match".
The Russian will face sixth-seeded Caroline Garcia of France in the third round.
Five-time major victor Sharapova leads the head-to-head against Garcia 4-1 but the French player won their only meeting in the last three years on clay in Stuttgart in April. In the quarterfinals, the American will face Anastaija Sevastova of Latvia, a 6-3, 7-6 (2) victor over 10th-seeded Julian Goerges of Germany.
Fifth seed and defending champion Elina Svitolina went through to the third round after in-form Romanian opponent Mihaela Buzarnescu was forced to retire with an ankle injury.
"Of course, it was not the plan to play just one match, but at the end, this is tennis".
German fourth seed Kerber wilted in the sticky heat in her first match since winning her third grand slam title.
Del Potro had a bye into the second round and was scheduled to face Dutchman Robin Haase.
After Suarez overcame a 5-1 deficit to tie the second set at 5, Stephens broke serve for a 6-5 lead and scored four straight points to put away the match, ending it with a sharp cross-court forehand.
The punishing serve that propelled Karolina Pliskova to the top of the tennis world a year ago proved ineffective against Kiki Bertens at the Rogers Cup in Montreal on Wednesday as the Dutchwoman cruised to a 6-2 6-2 victory.
Anastasija Sevastova of Latvia fought off a couple of match points to down Monica Puig 4-6, 7-6 (9), 6-1.