Top seed Cornet was beaten 7-6 (7-3) 6-3 by the Estonian qualifier in her first match since enjoying a strong showing at Wimbledon where she accounted for Serena Williams.
The French world number 21 is among five seeds to have bowed out of the competition after Anastasia Pavlyuchenkova fell on Monday.
An out-of-sorts Cornet double-faulted 13 times in the match after fourth seed Yaroslava Shvedova had also tasted defeat in the first round.
The Kazakh came unstuck against Germany's Laura Siegemund - losing 7-5 6-3 as the world number 201 booked herself a clash with Chanelle Scheepers in the WTA International clay-court event's second round.
The surprise results kept on coming as Anna Schmiedlova was beaten by Dutch qualifier Richel Hogenkamp in the final match while Polona Hercog crashed out to Germany's Dinah Pfizenmaier 7-5 7-6 (7-5).
Hogenkamp found herself 3-2 down in the first set but won 10 of the subsequent 11 games to progress with Pfizenmaier joining fellow German Annika Beck in the second round after the number six seed saw off Poland's Katarzyna Piter in three sets.
In a marathon encounter that saw 17 double faults in total, Beck came out on top 6-3 4-6 6-3 and will face Lara Arruabarrena next.
Alexandra Panova of Russia will take on one of the tournament's surprise packages Grace Min after seeing off Romanian Alexandra Cadantu in straight sets.
Min proved her credentials with a 6-0 6-4 win over second seed Pavlyuchenkova a day earlier and she will now meet Panova for the first time.
Brazil's Teliana Pereira withdrew due to injury at 3-1 down against Kazakh qualifier Yulia Putintseva while Estonia's Kaia Kanepi came through a difficult encounter with Sweden's Johanna Larsson.
World number 54 Kanepi will take on Siegemund next after winning through 6-2 4-6 7-5 in just over two hours.