Steven Gerrard's men came unstuck in the first leg of the third qualifying round tie, conceding two goals in the space of three second-half minutes.
Veljko Birmancevic was the star man for the hosts, who continued their momentum having come through the first two rounds of qualifying.
Birmancevic's right-wing cross was met with a brilliant 47th-minute volley from Soren Rieks to give the Swedish side the lead.
And two minutes later it was Birmancevic who doubled the advantage, slotting in off the post after pouncing on a defensive error from Connor Goldson.
A last-gasp deflected strike from Steven Davis gave Rangers hope for the second leg.
Yet, with the scrapping of the away goals rule, Rangers still need to score at least twice to guarantee progression to the playoff round.