World number two Petra Kvitova sealed the win with a 6-7 (2) 6-4 6-1 victory over Sabine Lisicki which gave the Czechs an unassailable 3-0 lead before the final two dead rubbers were shared in Stuttgart.
Last year's runners-up
Russian began Sunday's play with a 2-0 advantage but Carla Suarez Navarro put Spain back in the tie with a 6-0 6-3 victory over Nadia Petrova
However, Svetlana Kuznetsova sealed the Russians victory with a 6-2 4-6 6-3 success over Silvia Soler Espinosa, with Spain then taking the dead rubber doubles match.
Yanina Wickmayer had put the Belgians in charge with a 6-1 6-0 victory over Aleksandra Krunic but then Bojana Jovanovski levelled matter up with a 6-2 6-4 win over Kirsten Flipkens.
Jovanovski and Krunic then teamed up to defeat Wickmayer and Alison Van Uytvanck 7-6(2) 4-6 6-1 to seal the upset win.
Flavia Pennetta and Roberta Vinci won the topsy-turvy finale against Lesia Tusrenko and Olga Savchuk 7-5 0-6 6-1 to progress after Tsurenko had levelled the tie at 2-2 when a knee problem forced Errani to concede her singles match while trailing 6-1 3-0.
Francesca Schiavone had earlier had to dig deep to overcome Kateryna Bondarenko 6-7(6) 7-5 6-4.