The home team IFK Gothenburg opened the scoring through fullback Alexander Jallow.
Outfielder Hosam Aiesh found Jallow with a low post, which in turn found the net via a control on Mjällby defenders.
The goal came in the 42nd minute and stood for the half-time break.
But Mjällby withdrew purposefully.
After 73 minutes, Mjällby's Ivan Kricak sent a free kick in high over the penalty box.
IFK Gothenburg was centimeters from salvaging the victory.
Hussein Carneli slipped and slithered in the wet grass. His usual rapid runs were nowhere to be seen today.
But a few more goals never came and when the whistle sounded at the end of the match, it was scattered cries and bus whistles that came from a close on crowded Gamla Ullevi.
- We play against a good and cunning opponent.
What is sour is that we lead 1-0 for quite some time.
In the second half, I think we get better order in our defense and press game, however, we create far too little and have too bad a lead going forward, said IFK Gothenburg coach Mikael Stahre in Discovery's broadcast after the match.