Kvitova, who won Wimbledon in 2011, upended Jarmila Gajdosova in the opening match of the day 7-6 (7-2), 6-3.
Safarova had a tougher time against former U.S. Open champ Samantha Stosur but prevailed 7-6 (8-6), 7-6 (7-4).
Kvitova can advance the Czechs to the next round against either Italy or the United states if she beats Stosur in the first of Sunday's reverse singles matches.
If Stosur prevails, Safarova will try to win it for the Czechs by defeating Australia's Gajdosova.
Celebrity Families of 2014 [PHOTOS]