Gyor were first to create a gap, with Hatadou Sako's saves at one end and Bruna de Paula cutting through the Danish defence to open a 13-8 lead.
Esbjerg looked troubled, but Anna Kristensen's response between the posts fuelled an impressive run where the Danes scored five successive to goals to level the score and reignite the contest.
The teams remained deadlocked throughout most of the second half, as the momentum shifted frequently.
Esbjerg's defence kept Györ under pressure, while Kristensen continued to produce great reflex saves in goal.
The tension rose as the final minutes approached, with nerves on full display, which completely woke up Gyor's fans in the stands.
In a dramatic finish, Henny Reistad, who scored 10 on her return from injury, made a costly error, handing Gyor the chance to edge ahead.
Kristina Jorgensen scored, sealing an incredible win for the reigning champions, as Esbjerg failed to capitalise on their final four seconds.
