The 27-year-old's only appearance this year was at the French Open where she lost in the second round but she is hoping to hit form before Wimbledon in July.
Kvitova took one hour and 23 minutes to beat 22-year-old Smitkova, who is ranked 167th in the world.
The Czech will face Britain's Naomi Brody in the next round after she beat France's Alize Cornet.
Britain's world number seven Johanna Konta had to battle back from 5-2 down in the second set to beat Ukraine's Lesia Tsurenko in her opening match.
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Konta, runner-up at the Nottingham WTA event on Sunday, won the opening set 6-3 against world number 34 Tsurenko but the Ukrainian fought back well.
The British hope for a first women's Wimbledon title since Virginia Wade in 1977 was a game away from losing the second set but broke back at 5-3 and took the tiebreak 8-6.
Konta will face thee winner of the all-American match between Coco Vandeweghe and Christina McHale.
(Reporting by Simon Evans,; Editing by Ed Osmond)
