Cornet beat Yulia Putintseva of Kazakhstan 6-0, 6-0, and Lucic-Baroni overcame 11 double faults and held off Anett Kontaveit of Estonia 6-7, 7-6 (7-3), 6-3.
Cornet was added to the main draw, along with Katerina Siniakova, after injuries knocked out Timea Bacsinszsky and Samantha Stosur. Cornet was a semifinalist at the 2008 Connecticut Open and reached the Round of 16 at the French Open in June. She will face No. 2 seed Dominika Cibulkova in the second round Tuesday.
Defending champion and top-seeded Agnieszka Radwanska, the only player in the field ranked in the top 10, received a first-round bye. Three-time Connecticut Open champ Petra Kvitova is the No. 3 seed.
American Sloane Stephens withdrew Sunday due to a wrist injury. She has been replaced by Christina McHale, who lost a third-round qualifying match to Jana Cepelova.