Playing in front of a home crowd for the first time in two years, the Czechs began the highly-anticipated matchup with Safarova winning a first-set tiebreak against a struggling Francesca Schiavone, who was hobbled by a back injury.
Safarova built on that to breeze to a 6-1 win in the second set.
"I was hoping to win the (first) set because I had a break 5-4 serving for the set," she said. "I really didn't want to lose that set, but I turned it and was able to win in the tiebreak so it was amazing."
World number three Kvitova took less time to power past Italy's Sara Errani, beating her 6-4 6-3 in 81 minutes.
The Czechs and Italians have won the last three Fed Cups, with the Czech Republic beating Russia last year for their first title since they won in 1988 as Czechoslovakia.
Italy have won the past four head-to-head meetings.
Kvitova will play Schiavone on Sunday, while Safarova is slated to play Errani.
Andrea Hlavackova and Lucie Hradecka are scheduled to play Italy's Errani and Roberta Vinci in the doubles.
The winners of the tie will take on the winners of the Russia-Serbia match-up, which is poised at 1-1 after the first two rubbers.
After Russian number two Anastasia Pavlyuchenkova was beaten by Jelena Jankovic 6-4 6-3 in the opening singles rubber, Kuznetsova prevailed over Ivanovic 6-2 2-6 6-4 to even the two-day tie for the hosts.
"Ana always is a difficult opponent for me," former world number two Kuznetsova, who has slipped to number 27 after struggling to regain her top form this season, told reporters.
"But I think this court was suiting my game better than her," the Russian said about the red clay court, laid on top of the ice rink at Moscow's Megasport arena.
"I know my team relies on me to win my matches and I will do whatever it takes to keep us in this tie."
The Russians, bidding for their fifth Fed Cup title since 2004, have won their three previous ties against Serbia.
Former world number one Jankovic was far too consistent for error-prone Pavlyuchenkova, who blamed her poor service games for losing to the Serb for the first time in four meetings.
"My serving was just terrible today, a real nightmare," said the 20-year-old Russian.
Jankovic said: "I was a bit nervous early on, but then managed to take control of the match. It was a very important win for me because I was playing for my country."
Jankovic will aim to put Serbia, playing in their first Fed Cup semi-final, ahead again when she takes on Kuznetsova in the first reverse singles on Sunday while Ivanovic faces Pavlyuchenkova in the second match.