Cornet cruised past Moroccan wildcard Fatima Zahrae El Allami 6-1, 6-1 while fifth-seeded Bertens rallied from a 5-2 deficit in the first set to beat Hungarian qualifier Timea Babos 7-5, 6-2.
Two-time runner-up Simona Halep of Romania, Kristina Mladenovic of France, Tsvetana Pironkova of Bulgaria and Kaia Kanepi of Estonia also won their first-round matches.
Pironkova, a 2010 Wimbledon semi-finalist, broke twice in each set to dismiss Karolina Pliskova of the Czech Republic 7-5, 6-1, while the fourth-seeded Kanepi knocked out 2009 champion Anabel Medina Garrigues of Spain 4-6, 6-3, 6-2.
Halep dominated Nina Bratchikova of Russia 6-1, 6-3, and the seventh-seeded Mladenovic edged Alexandra Cadantu of Romania 6-4, 4-6, 6-4 in a match with 13 breaks of serve.
Cadantu replaced Dominika Cibulkova in the main draw after the top-seeded Slovakian withdrew because of an injury.
Mandy Minella of Luxembourg, Italian qualifier Karin Knapp and Spaniards Silvia Soler-Espinosa and Lourdes Dominguez Lino also moved into the second round.
Dominguez Lino jumped to 4-0 in the final set to oust Misaki Doi of Japan 6-2, 4-6, 6-2 while Soler-Espinosa saved eight break points to defeat Slovakian qualifier Michaela Honcova 7-6 (7/4), 6-2.
Knapp won the first 11 games in her 6-0, 6-1 rout of Moroccan wildcard Lina Qostal and Minella outlasted Spanish qualifier Estrella Cabeza Candela 7-6 (7/3), 4-6, 7-5.
Results on the second day of the WTA Princess Lalla Meryem tournament at Marrakech, Morocco, on Tuesday (x denotes seeding):
Women's singles
First round
Kaia Kanepi (EST x4) bt Anabel Medina Garrigues (ESP) 4-6, 6-3, 6-2
Tsvetana Pironkova (BUL x9) bt Karolina Pliskova (CZE) 7-5, 6-1
Lourdes Domnguez (ESP) bt Misaki Doi (JPN) 6-2, 4-6, 6-2
Kiki Bertens (NED x5) bt Timea Babos (HUN) 7-5, 6-2
Mandy Minella (LUX) bt Estrella Cabeza (ESP) 7-6 (7/3), 4-6, 7-5
Silvia Soler (ESP) bt Michaela Honcova (SVK) 7-6 (7/4), 6-2
Simona Halep (ROM) bt Nina Bratchikova (RUS) 6-1, 6-3
Karin Knapp (ITA) bt Lina Qostal (MAR) 6-0, 6-1
Alize Cornet (FRA x3) bt Fatima El Allami (MAR) 6-1, 6-1
Kristina Mladenovic (FRA x7) bt Alexandra Cadantu (ROM) 6-4, 4-6, 6-4