Kvitova rallied from poor first set to overcome Errani 2-6, 6-2, 6-0 after a rain delay at Palermo, Italy, staving off a sweep after the Italian squad had grabbed a 2-0 lead in Saturday's singles events.
Errani seemed on her way to a dominating victory after handling Kvitova in just 38 minutes in the first set, but the Czech bounced back, taking the final 11 games after trailing 2-1 in the second set.
More inclement weather after the event prompted officials to delay the best-of-five tie's final two events to Monday.
Roberta Vinci, who bounced Kvitova in straight sets Saturday, will try to wrap up an Italian victory against the Czech Republic's Lucie Safarova in Monday's second reverse single match.
Should Safarova win, the semifinal will be decided by a doubles match.
The winner will advance to November's final to meet Russia, which prevailed in a comeback victory over Slovakia in the weekend's other semifinal.