Voegele failed to serve out the second set at 5-3 as Zakopalova leveled at 5-5 but the 58th-ranked Swiss player didn't concede a game afterward. Zakopalova lost her serve eight times and saved 19 more break points.
The result ended Voegele's seven-match winless streak on the WTA tour, which started in the second round at Stanford in July.
The 44th-ranked Svitolina defeated Mona Barthel 6-2, 7-5 after the German double-faulted eight times and dropped five service games.
Schiavone hit four aces in beating Karolina Pliskova 7-5, 6-2.