Cohen-Aloro saved six break chances against her in the first set and took advantage of Cornet's ineffective service game - which netted just seven points in the opening set - to break her 14th-ranked opponent three times.
The 97th-ranked Cohen-Aloro then broke Cornet at the first opportunity in the second game of the second set for a 2-0 lead and then hung on for the victory after both players traded break points.
Second-seeded Flavia Pennetta of Italy broke Petra Kvitova seven times to overcome four breaks of her own serve in a 6-3, 6-3 win over the Czech. Pennetta will play Roberta Vinci, who beat Pauline Parmentier of France 6-4, 7-6 (7), in the second round.
Nearly a week after upsetting top-ranked Serena Williams in Andalucia, Klara Zakopalova lost 6-1, 7-6 (7) to Slovakia's Magdalena Rybarikova to bow out of the tournament.
In other first round matches, seventh-seeded Sara Errani of Italy rallied for a 2-6, 7-5, 6-3 victory against Mathilde Johansson and Lucie Safarova of the Czech Republic beat Tsvetana Pironkova 6-1, 3-6, 6-0.
Russia's Maria Kirilenko, seeded sixth, begins her title defense against Camille Pin of France on Tuesday.
