Simon Lundevall struck the only goal of the tie early in extra time as Elfsborg defeated 10-man Randers 1-0 to reach the UEFA Europa League third qualifying round.
Following a goalless first leg in Denmark last week, the two sides could not be separated after another 90 minutes at the Boras Arena.
Elfsborg's Viktor Claessen came closest to winning it in normal time, striking the post shortly after the break.
Randers then saw striker Nicolai Brock-Madsen sent off for a second caution in the 84th minute for leading with his arm in an aerial challenge with Sebastian Holmen.
The incident drew blood from the head of Holmen, who was faced with an irate Randers coach Colin Todd on the sideline as he received treatment.
And Todd's mood will not have been improved four minutes into extra time as Lundevall shot home from 20 yards, booking Elfsborg a meeting with either Shamrock Rovers or Odd.
‘He instantly popped into our ratings as one of the best U21 midfielders in Europe. The impressive thing is Slot has made him into more of a no.6' Inside Ryan Gravenberch's transformation at Liverpool
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
