Vinci, the winner of nine WTA singles titles, overcame a challenge from Alexandra Panova, who has reached only one WTA final. Vinci rallied from 5-2 down in the second set and saved four match points to win 5-7, 7-5, 8-6 in 3 hours, 13 minutes.
Errani, playing in her second Fed Cup final, overcame Fed Cup newcomer Irina Khromacheva 6-1, 6-4.
Russia has won the tournament four times in the past 10 years. But the Russians are without any of its top players, with some injured and others playing in the season-ending Tournament of Champions in Bulgaria.
Italy needs one more victory on Sunday to add to its Fed Cup titles in 2006, 2009 and 2010.
The reverse singles are Sunday with Errani taking on Panova and Vinci facing Khromacheva.
Vinci and Errani are the No. 1-ranked doubles team in the world. But Italy captain Corrado Barazzutti has chosen Flavia Pennetta -- who has been a part of three Fed Cup-winning teams -- and Knapp to face Kleybanova and Gasparyan.