In the first rubber, Kvitova, the Czech number one and world number eight, beat Australia's 168-ranked Jarmila Gajdosova 7-6(2), 6-3 in an hour and 35 minutes.
Two tie-breaks were required for Safarova, the world number 18, as she won 7-6(6), 7-6(4) against Samantha Stosur in two hours and 28 minutes.
Kvitova, the 2011 Wimbledon champion who led the Czechs to Fed Cup victories in 2011 and 2012, continues to thrive in the Cup for her country.
"It wasn't an easy match and she had moments when she played very well, but she also helped me a few times," the 22-year-old Kvitova said. "I was terribly nervous in the first set."
On Sunday, Kvitova will first face Stosur. If she gives the Czechs a winning 3-0 lead, the tie will go straight to doubles, while if she loses, Safarova will take on Gajdosova.
In the doubles, Czechs Andrea Hlavackova and Lucie Hradecka, the world numbers three and four in the format, will face Australian Open finalists Casey Dellacqua and 16-year-old Ashleigh Barty.
World Group results
Italy 1-1 USA
Varvara Lepchenko (U.S.) beat Roberta Vinci (Italy) 2-6 6-4 7-5
Sara Errani (Italy) beat Jamie Hampton (U.S.) 6-2 6-1
Czech Republic 2-0 Australia
Lucie Safarova (Czech Republic) beat Samantha Stosur (Australia) 7-6(6) 7-6(4)
Petra Kvitova (Czech Republic) beat Jarmila Gajdosova (Australia) 7-6(2) 6-3
Serbia 1-1 Slovakia
Vesna Dolonc (Serbia) beat Dominika Cibulkova (Slovakia) 4-6 4-5 (Cibulkova retired)
Daniela Hantuchova (Slovakia) beat Bojana Jovanovski (Serbia) 7-5 6-2
Russia 1-1 Japan
Ayumi Morita (Japan) beat Ekaterina Makarova (Russia) 6-2 6-2
Maria Kirilenko (Russia) beat Kimiko Date-Krumm (Japan) 7-6(3) 6-4(0)