Throughout the match it otherwise looked like Brondby would head home to Denmark with a crucial win in the bag. On the artificial grass in the Faroese capital, the visitors dominated throughout the 90 minutes, but struggled to make their dominance tell in the decisive moments.
Striker Filip Bundgaard has had difficulty getting playing time since arriving at Brondby, but after the sale of Mathias Kvistgaarden and Yuito Susuki, Bundgaard has seized the chance to shine.
On Sunday he scored twice against Silkeborg, and on Thursday evening he also found the back of the net to open the scoring for the Danes.
After an hour's play, Nicolai Vallys split the Torshavn defense apart with a deep pass, and Clement Bischoff had no trouble picking out the onrushing Bundgaard, who was left with an easy task to push the ball into the open goal.
Until then Brondby had been gulty of wasting huge chances in front of goal.
A minute after the break, everything was set for the new full-back Oliver Villadsen to score his first goal in the Brondby shirt, but in the most incredible way he managed to divert the ball over the bar from close range after a perfect cross from the left.
Just a minute later, Bundgaard received a perfect delivery in front of goal but with the goal at his mercy, his bullet header went over the bar.
Despite breaking the deadlock Brondby failed to make the most of their momentum.
Instead, the home team began producing a series of dangerous attempts and were unfortunate to see Emil Pedersen's effort saved by the woodwork.
It still looked like a narrow Danish win until Jákup Thomsen beat Brondby keeper Patrick Pentz with a shot from the edge of the box shortly before the end.
Despite the setback, the Danish favorites still have great chances of securing their ticket for the third round in the return leg at home next Thursday.
Should they be successful, Brondby will face the winner of the clash between Albanian outfit KF Vllaznia and Vikingur Reykjavik from Iceland in the third round. The Albanians won the first meeting at home on Thursday with 2-1.