Zverev, who recently claimed his first Grand Slam title at the French Open, looked in control as he took the opening set comfortably. But Kopriva, ranked 64th, hit back to level the match, as he won a late break point and then claimed the second set, pushing the match to a decider.
Kopriva finished the second set with 14 winners, while Zverev finished with 11, before the German regrouped and closed out a 6-3, 4-6, 6-2 victory to start his grass season with a win.
That sets up an all-German clash with Yannick Hanfmann in the next round. He defeated rising Brazilian teenager Joao Fonseca 6-2, 6-2 in the first round.
Earlier, Italian qualifier Mattia Bellucci produced a clinical display to knock out Bublik 7-6(6), 6-1 in just 75 minutes. The world number 74 outplayed the seventh-seeded Kazakh on his preferred grass surface.
Bublik, a champion in Halle in 2023 and 2025, also missed an opportunity to strengthen his push towards the top 10 following the defeat.
Bellucci will next face Belgium’s Raphael Collignon for a place in the quarter-finals.