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Teichmann produced an astonishing fightback to beat No.4 seed Svitolina 2-6 6-4 7-6 (7-5) at the Caja Magica.
Unseeded Swiss Teichmann came from a set and a break down to claim the scalp of the world No.5, who squandered a commanding 5-1 lead in the final set.
Svitolina saw six match points come and go in an incredible deciding set, opportunities she was left to rue when the 40-ranked Teichmann won a tie-break to end a contest that took two hours and 33 minutes to settle.
World No.1 Barty extended her winning run on clay to 12 matches with a 6-2 6-1 success over Shelby Rogers.
Barty claimed her third title of the year in Stuttgart last weekend and the Australian needed just an hour to send Rogers packing, losing just two points on her first serve and not facing a single break point.
French Open champion Iga Swiatek was also an emphatic winner, easing past another American in the form of Alison Riske 6-1 6-1.
Swiatek, making her debut in this tournament, has won 16 consecutive matches on clay after blowing Riske away.
Defending champion Kiki Bertens beat 15-year-old Victoria Jimenez Kasintseva 6-4 6-0, while Petra Kvitova progressed when opponent Marie Bouzkova retired because of a hand injury after the Czech had won the first set.
Garbine Muguruza was forced to withdraw ahead of her meeting with Sloane Stephens with a leg injury. Belinda Bencic, Angelique Kerber and Johanna Konta were among the other winners in the Spanish capital.