Cornet, ranked 21st, had eight double faults and dropped serve six times to exit the tournament the day after top-seeded Andrea Petkovic of Germany lost her opener in straight sets.
The 73rd-ranked Bertens faces either eighth-seeded Monica Niculescu of Romania or Mona Barthel of Germany in the quarterfinals. They played late Wednesday.
Zahlavova Strycova won 6-4, 7-6 (1) against Belgium's Alison Van Uytvanck in a match featuring 11 service breaks, while Lepchenko saved all eight break points she faced in a 6-2, 6-2 win against Slovakia's Jana Cepelova.
Zahlavova Strycova next plays qualifier Johanna Larsson of Sweden, who rallied past Polona Hercog of Slovenia 6-7 (8), 7-5, 6-1, while Lepchenko's quarterfinal opponent will be either No. 3 seed Sabine Lisicki of Germany or Denisa Allertova of the Czech Republic, who played late.
Sixth-seeded Roberta Vinci of Italy also went out in the second round after losing 7-5, 6-0 against Annika Beck of Germany, who broke her serve seven times.
Beck next plays Austria's Patricia Mayr-Achleitner, who won 6-1, 7-5 against France's Pauline Parmentier.