World number 16 Stephens joins fellow seed 2012 winner Cornet in falling at the first hurdle after Kirsten Flipkens and Elena Vesnina bowed out a day earlier.
The American fell to Julia Georges, who converted three of five break points in the match to win the pair's first meeting on clay and secure her second top-20 scalp of the year, having beaten Sara Errani in the first round of the Australian Open.
Cornet was dumped out by Italy's world number 51 Camila Giorgi thanks to a 4-6 6-1 3-6 defeat.
Having produced an impressive second-set showing, she crucially dropped serve late in the match, allowing Giorgi to serve out for the match.
Sixth-seed Bojana Jovanovski was another seed to tumble in Strasbourg as she was beaten 6-4 6-0 by Croat qualifier Mirjana Lucic-Baroni.
The world number 107 won the last seven games of the match to send Jovanovski out with Andrea Petkovic and Shuai Peng the only seeds to progress on the day.
Former winner Petkovic was too strong for France's Pauline Parmentier while Peng ensured she was not a casualty despite being given a stern examination by Australian qualifier Ashleigh Barty.
The number seven seed won 80 per cent of her first service points to see off a stubborn Barty and set up a second-round meeting with Silvia Soler-Espinosa.
Stefanie Voegele's resistance proved too little, too late against the Spaniard as she was unable to recover from a woeful first set.
The world number 118 got off to a flyer against the 24-year-old Swiss - moving ahead after just 22 minutes - before wrapping up the match 6-0 7-6 (7-5).
Christina McHale and Alexandra Cadantu are separated by 21 places in the rankings and their first meeting went to form as the American was a convincing 6-1 6-2 winner.
McHale broke six times in the match as she comfortably progressed into the second round, while compatriot Lauren Davis is also safely through.
She saw off Belarusian qualifier Olga Govortsova 6-2 6-4 and will take on Goerges next looking to reach the last eight for the third time this year.
Monica Puig was made to work hard in her opener as she moved past Johanna Larsson of Sweden 7-5 7-6 (7-5).