Cornet rolled to a 6-4, 6-1 victory over fourth-seeded Angelique Kerber at IGA Stadium, and Bertens was just as sharp while producing a 6-2, 6-2 triumph over ninth-seeded Karolina Pliskova of the Czech Republic.
It marked the second straight time that Cornet prevailed against Kerber after losing their first three matchups.
"Last year in Beijing, I was more focused about playing the forehand, moving her," the Frenchwoman told reporters, while referencing her first win over Kerber. "It was a different tactic today. It was more like playing on the backhand side, also being aggressive.
"It was different, and I'm happy I actually adapted my tactic depending on her, and that I could do it the whole match."
Kerber committed 32 unforced errors -- including 20 in the opening set -- while going down meekly.
"It was my first match after three weeks now," the German said. "It is always not so easy to change also the surface. I think I need a little bit more time to get used to hard court again, to have a lot of matches, a lot of practice sessions on hard."
While the Wimbledon champ contemplates what needs fixing, Cornet will face 15th-seeded Ashleigh Barty of Australia in the third round. Barty moved on with a 7-6 (7), 6-2 win over Belgium's Alison Van Uytvanck.
Meanwhile, Bertens needed just 85 minutes to finish off Pliskova, who committed eight double faults and never found her serving rhythm.
"Happy with my 2nd round win," Bertens tweeted after the victory.
Bertens, from Netherlands, will attempt to record another upset in the third round when she faces eighth-seeded Czech Petra Kvitova.
Also, former World No. 1 Maria Sharapova dispatched 12th-seeded Daria Kasatkina 6-0, 6-2 in an all-Russian affair.
Sharapova required just 76 minutes to complete the sweep. She recorded 25 winners while Kasatkina hit just three.
"I came into the match with a lot of respect for my opponent," Sharapova said afterward. "She's had really good results. She's No. 12 in the world. You can't take that very lightly, especially for a second round. I thought, yeah, with everything, I took that and I made good use of it."
Fifth-seeded Ukrainian Elina Svitolina moved on with a 6-3, 6-7 (5), 4-3 victory over Romania's Mihaela Buzarnescu, a match that was halted when Buzarnescu couldn't continue due to a right ankle injury. Buzarnescu was removed from the court in a wheelchair with tears streaming down her face.
Among other matches, 14th-seeded Belgian Elise Mertens defeated China's Zhang Shuai 6-2, 6-7 (2), 6-0, while third-seeded American Sloane Stephens rolled to a 6-0, 6-2 victory over Canadian Francoise Abanda.
--Field Level Media