The Czech left-hander was the last winner of the German tournament, beating Anett Kontaveit in the 2019 final, with last year's event cancelled due to the pandemic.
Kvitova was handed a tough opening assignment against Australian Open runner-up Brady, but the American's streak of defeats since that Melbourne final defeat extended to three as Brady lost 6-4 6-3 in an hour and 32 minutes.
Two-time former Wimbledon champion Kvitova, seeded seventh, picked apart the Brady serve once in each set and fended off three break points across the contest to secure her last-16 place.
She was joined in the second round by powerful Belarusian Aryna Sabalenka, as the fifth seed scorched to a 6-2 6-2 win over China's Zhang Shuai.
Earlier, Ekaterina Alexandrova, Kontaveit, Anna Lena Friedsam and Laura Siegemund also went through to the last 16 of the WTA 500 event.
Meanwhile, Fiona Ferro caused a shock at the Istanbul Cup as she landed a 7-5 6-2 first-round win over former champion Petra Martic.
Martic, seeded second this year and ranked 21st in the world, landed the title in Istanbul in 2019 but was rolled over by French player Ferro in an hour and 45 minutes, in a match featuring seven breaks of serve.
Veronika Kudermetova followed her recent maiden title in Charleston with a nerve-jangling first-round win against Bernarda Pera, the fifth seed scrambling through 6-4 3-6 7-6 (7-4).
There was no such toil for top seed Elise Mertens, a 6-2 6-0 winner against Lara Arruabarrena, or for Daria Kasatkina, who dished out a 6-1 6-0 drubbing to Tereza Mrdeza.
Kasatkina is seeking her third title of 2021, having won previously in Melbourne and St Petersburg.
Anastasia Pavlyuchenkova bowed out of the WTA 250 event though, beaten 7-6 (7-2) 6-7 (8-10) 6-4 by compatriot Anastasia Potapova.