Djurgarden could have been undone early on by Pafos other Sweden international, but Ken Sema could not keep his header down in the second minute.
The home side maintained the upper hand through the first quarter of the match, with Jairo and Bruno both getting shots on goal, only to be thwarted by visiting goalkeeper Malkolm Nilsson on both occasions.
But Djurgarden gained a foothold on the game around the 30-minute mark - a driving run from Tokmac Chol Nguen ended with him pulling his strike narrowly wide, before Oskar Fallenius and Miro Tenho tested Ivica Ivusic in the Pafos goal.
Failure to take one of those chances would have been punished by Pepe late in the half, if it was not for another smart Nilsson save, meaning the teams went in goalless at the break.
The second half in Limassol failed to ignite until just before the hour mark, when Nguen sent a free-kick whistling past the post.
But after 65 minutes, the deadlock was finally broken and it was the five-time Swedish international, Tankovic, who scored it.
A hopeful ball into the box was not dealt with by the Djurgarden defence, allowing Vlad Dragomir to flick it across goal and into the path of Tankovic, who bundled home his first goal of the competition proper.
Tankovic thought he had doubled his and his team's tally four minutes on, only for his neat finish to be ruled out by VAR for offside.
But one goal turned out to be enough, as Jani Honkavaara's side failed to register a shot on target in the second half.
The disappointing result for the Iron Stoves sees their four-match winning run in the competition come to an end. Meanwhile, Pafos continue to impress on their first European campaign.
The tie will conclude next Thursday, March 13th, with the second leg at Djurgarden's 3Arena.