The 23-year-old Kostyuk won 6-3, 6-4 in the first WTA final between two Ukrainian players.
She succeeded countrywoman Elina Svitolina, who won in Rouen last year as the top seed.
For 19-year-old qualifier Podrez, ranked 209th in the world, it was a first final in her main draw debut at a tour-level event.
Both Kostyuk and Podrez struggled on their serve in the indoor clay final, hitting 12 double faults between them.
But Kostyuk converted six of her nine break point opportunities compared to Podrez, who converted four of seven in the tie, which lasted 1hr 12min.
Kostyuk claimed her second WTA singles title and first in more than three years since her victory in Austin, Texas, in 2023.