In the first half of the regular time, Silkeborg was heading for a big disappointment with the home side leading 1-0 at the break.
Silkeborg turned things around in the second half, when Tonni Adamsen got revenge for a missed penalty and scored twice within eight minutes. Akureyri then managed to restore parity forcing the game to be decided in extra time.
Akureyri were the better team for large periods in extra time, but in the 114th minute, Tonni Adamsen thundered the ball into the net from the edge of the box via the crossbar to complete his hat trick.
At the start of the game, it didn't seem like the underrated Icelandic outfit would cause Silkeborg any problems.
The visitors dominated, and after just a few minutes, they were awarded a penalty when the ball grazed the hand of an Icelandic defender in box, but Tonni Adamsen's weak attempt allowed Steinthor Audunsson to save the shot.
The home side were then more clinical when they won a penalty as Nicolai Larsen mistimed an intervention in the box and Hallgrimur Steingrimsson had no problems converting from the spot.
Just under ten minutes into the second half, Adamsen then atoned for his error when he equalized after a pin-point pass from Jens Martin Gammelby.
Then after an hour the visitors took the lead as Adamsen doubled his score of the night when he elegantly lifted the ball over an onrushing Akureyri keeper.
Silkeborg were content just to defend towards the end of the match which was punished by the home side six minutes before the end when Vidar Orn Kjartansson made it 2-2 to sent the match into extra time where Adamsen stole the win for the Danes.
In the next round Silkeborg will face Polish outfit Jagollonia.