The Swedish side were surprisingly held to a goalless draw by the Luxembourg outfit in the first leg and they left it late to progress at Stade Josy Barthel on Thursday.
Extra time was looming until Danish striker Lasse Vibe broke the deadlock 14 minutes from time and May Mahlangu eased the two-time UEFA Cup winners' nerves with a second goal nine minutes from time.
Mikael Stahre's men breathed a sigh of relief after making hard work of progressing and now face Gyori ETO in the second qualifying round.
Adam Rooney scored the first hat-trick of the competition as Aberdeen beat FK Daugava Riga 3-0 to seal an emphatic 8-0 aggregate victory and set up a clash with Eredivisie side FC Groningen.
The striker also became the first Aberdeen player in 30 years to score a treble in European competition, with Scotland assistant manager Mark McGhee the last player to achieve that feat.
There was late drama at the ISS Stadion in Vantaa, where hosts Honka were left stunned as JK Sillamae Kalev scored in the last minute of extra time to put the visitors from Estonia through on away goals after losing 3-2 on the night.
Sillamae led 2-1 after the first leg and increased their advantage when Andrei Sidorenkov put them in front on the night, but Ilari Aijala equalised on the stroke of half-time and Kevin Mombilo struck right at the death to force extra time.
Roni Porokara put the home side in front and the fightback looked to be complete until Evgeni Kabaev had the final say in the last minute of extra time.
It was not only the Finnish side who suffered late heartbreak, as Maksim Maksimov struck with just a minute remaining to ensure Atlantas beat Differdange 03 3-1 to secure a 3-2 aggregate victory.
Luxembourg outfit Differdange led 1-0 from the first leg, but Mehdi Martin's red card after only 19 minutes left them up against it in the second leg after Kazimieras Gnedojus had levelled the tie.
Marius Papsys' penalty put the home side in front on aggregate, but Differdange refused to lie down and Jean-Philippe Caillet levelled the tie nine minutes from time, only for Maksimov to settle it right at the death.
KF Laci held their nerve to beat Rudar Velenje 3-2 on penalties after the two sides had played out another 1-1 draw, while the tie between Litex Lovech and Veris was postponed due to a waterlogged pitch.
Ferencvaros overcame a first-leg deficit to beat Sliema Wanderers 2-1 - and 3-2 on aggregate - while Rosenborg, Shakhter Karagandy, Astana, RNK Split, Víkingur Gota and Buducnost Podgorica also progressed.
Crusaders also went through on Thursday, along with Zeljeznicar Sarajevo, Chikhura Sachkhere, FK Haugesund, Inter Baku, Nomme Kalju, Cukaricki, Zimbru Chisinau and Flamurtari Vlore.
IF Brommapojkarna can also look forward to a second round qualifying clash, as can Tromso, who hammered Tartu FC Santos 13-1 on aggregate, Diosgyori VTK, Kairat Almat, Spartak Trnava, Dery City and UMF Stjarnan.
FC Vaduz's European adventure will continue, while MYPA, Sligo Rovers, Botev Plovdiv, Luka Koper, NK Siroki Brijeg, Dundalk, FH Hafnarfjordur and Metalurg Skopje live to fight another day.
