The pair broke each other's serve twice in the first set before the 161st-ranked Kontaveit won the tiebreaker against the 21st-ranked Cornet. The 18-year-old Kontaveit next faces Jana Cepelova of Slovakia.
Cornet wasn't the only seeded player to go out on Tuesday. Laura Siegemund of Germany defeated fourth-seeded Yaroslava Shvedova of Kazakhstan 7-5, 6-3, while Richel Hogenkamp of the Netherlands beat fifth-seeded Slovakian Anna Schmiedlova 6-3, 6-1.
Also, Germany's Dina Pfizenmaier advanced to the second round by defeating two-time winner and eighth-seeded Polona Hercog of Slovenia 7-5, 7-6 (5).
However, sixth-seeded Annika Beck of Germany beat Poland's Katarzyna Piter 6-3, 4-6, 6-3 after breaking her serve 10 times.