The 22-time champions of Norway had home advantage in the second-round qualifying tie at the Lerkendal Stadion.
However, they will travel to Ireland needing to overturn the one-goal deficit, as second-half goals from Alan Keane and Danny North put the visitors in control.
Mikkel Diskerud offered hope of a late comeback when he struck nine minutes from time, but it proved in vain as Sligo held on for a precious win.
Koper were also 2-1 victors on their visit to Neftci, while the likes of Botev Plovdiv, Mlada Boleslav, Split and Zulte Waregem still have to work to do after winning by the same scoreline at home.
IFK Gothenburg have won Europe's secondary club competition on two occasions in 1982 and 1987, and the Swedish outfit look destined for the next round after a 3-0 win at Gyori ETO.
Lasse Vibe got the visitors off to a flying start in the eighth minute and Jakob Johansson and May Mahlangu added furth goals in the second half.
Fellow Swedish sides Brommapojkarna and Elfsborg had differing fortunes. The former routed Crusaders 4-0, while the latter were beaten 1-0 at home by Inter Baku.
Russian Premier League outfit Krasnodar are virtually assured a spot in the third round of qualifying thanks for a 4-0 win at Sillamae Kalev.
Ari, Yury Gazinskiy, Wanderson and Vladimir Bystrov were all on the scoresheet in the comfortable win.
Grodig too won 4-0 on their travels at Cukaricki, and Trencin secured the same score at home to Vojvodina. Molde put one foot in the next round thanks to a 4-1 home success over Gorica.
Kazakh outfit Astana should progress after a 3-0 home win over Hapoel Tel Aviv, while Belarus side Dinamo Minsk beat MYPA by the same score. Zorya won 3-0 on their visit to Laci.
The matches between Kairat and Esbjerg, RoPS and Asteras Tripolis, CSKA Sofia and Zimbru, Luzern and St. Johnstone and Neman Grodno and FH are all nicely poised after finishing 1-1.
Shakhter Karagandy, Spartak Trnava and Tromso played out stalemates on their visits to Atlantas, Zestafoni and Vikingur, while Jagodina took a valuable 0-0 draw away from CFR Cluj.
There were no goals in the matches involving Bursaspor and Chikhura Sheva, Aberdeen and Groningen, or Metalurg Skopje and Zeljeznicar.
Omonia Nicosia and Diosgyor take 2-0 leads back to their respective home legs with Buducnost and Lovech. Petrolul Ploiesti scored twice in the last 13 minutes to win 2-0 at home to Flamurtari Vlore.
Dundalk face a tricky task after losing 2-0 at home to Hajduk Split, while Motherwell surrendered a two-goal lead in a 2-2 draw at home to Stjarnan.
One goal was all that was needed for Slovan Liberec to win on their visit at Kosice, while Nomme Kalju were 1-0 victors at home to Lech Poznan. Ruch Chorzow also have a one-goal cushion ahead of the second leg following an entertaining 3-2 triumph over Vaduz.
Linfield and Rijeka triumphed 1-0 in their home ties, as did Shakhtyor and Haugesund on their visits to Derry City and Sarajevo.
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