Kristianstad kept tabs on their experienced opponents in the opening minutes, but from 10-8 up, Szeged put together a 6-1 run that kept them in full control of proceedings from then on, before another 4-0 run saw them pass the 20-goal mark before half-time, 22-13.
The visitors were having little trouble finding space in the Szeged defence, only to regularly come unstuck against the hosts' new goalkeeper, Nikola Portner, whose eight saves were crucial to victory.
In the second half, Szeged took the margin into double figures, and a best run of 4-0 extended the advantage to a game-high 12 goals, as Magnus Rod top scored with nine goals from just 10 shots, while 12 players got on the scoresheet for the home side.
Anton Hallback scored the most for the Swedish champions with five, not nearly enough as Szeged took a comfortable 42-31 victory, as the former quarter-finalists surpassed 40 goals in a Champions League match for the first time.
Szeged sit top of the group after one round, though they are not the highest scorers among the four teams, as Sporting CP beat GOG 43-33. Kristianstad have the daunting task of welcoming Sporting in round two.