Keys pulled off a 6-2, 6-4 win over Cornet in the first round of the Western & Southern Open on Monday.
''I don't have many complaints about today,'' the 19-year-old American said. ''I definitely played pretty well and stayed in my game for the most part. There's obviously a couple of things that I still want to work on and have better for next round, but I'm pretty happy with today.''
Cornet, who upset the top-ranked Williams in the third round at Wimbledon six weeks ago, was playing with an injured left leg that she said started bothering her during her second-round loss to Victoria Azarenka last Tuesday at the Rogers Cup in Montreal.
''That's not the reason I lost, but it didn't help,'' said the Frenchwoman, who lost to Keys earlier this year. ''She was very aggressive, hitting all winners.''
Three qualifiers advanced in other women's first-round action. Taylor Townsend outlasted Klara Koukalova, 6-3, 4-6, 7-5, while Pauline Parmentier eliminated Casey Dellacqua, 7-6 (4), 6-2, and Karin Knapp upset wild-card Belinda Bencic, 6-2, 7-6 (1).
Also, Kirsten Flipkens defeated qualifier Polona Hercog, 6-3, 6-2, while Zhang Shuai beat Heather Watson, 6-3, 2-6, 7-5, and Elina Svitolina cruised past Lauren Davis, 6-3, 6-2.
Fernando Verdasco needed three sets to eliminate Marcel Granollers, 5-7, 6-3, 7-6 (5). Three other players advanced with first career wins over their opponents, led by qualifier Benjamin Becker, whose 7-6 (5), 6-4 upset of Ivo Karlovic was the German's first win in seven matches against the 28th-ranked Croatian. Also, Philipp Kohlschreiber beat Jeremy Chardy, 6-3, 4-6, 6-4, the 24th-ranked German's first win in three matches against the Frenchman, and Lu Yen-Hsun advanced with a 6-3, 6-4 win Denis Istomin, Lu's first win in four matches against Istomin.
Qualifier Marinko Matosevic moved on with a 6-4, 7-6 (4) win over Nicolas Mahut, while Gael Monfils needed three sets to get past 63rd-ranked Federico Delbonis, 6-3, 3-6, 6-3.