Trailing 2-1 from the first leg in Norway, the former UEFA Champions League quarter-finalists fell further behind to Danny North's 13th-minute opener.
That left the visitors needing three unanswered goals to progress, the first of which arrived just three minutes later through Pal Helland and a second-half brace from Mike Jensen saw the Norwegian outfit advance to a tie with Karabukspor in the next round.
Elfsborg's 1-0 win at Inter Baku forced a penalty shoot-out, won 4-3 by the Swedish outfit, while Georgian side Chikura also prevailed on spot-kicks after a goalless draw with Bursaspor.
Krasnodar's 5-0 thumping of Sillamae Kalev completed a 9-0 aggregate rout, and Shakhtyor's 5-1 win over Derry City also saw them through in comfortable fashion.
Trencin almost threw away a 4-0 first-leg advantage against Vojvodina, losing 3-0 in Serbia to squeeze through to a clash with Hull City, but there were no such problems for Brommapojkarna, who drew 1-1 with Crusaders to progress 5-1 and set up a meeting with Torino.
Goalless draws were good enough for the likes of Dinamo Minsk, Omonia Nicosia, Ruch Chorzow and Split to make it through, while CSKA Sofia's stalemate with Zimbru saw them crash out on away goals after a 1-1 draw in Bulgaria in the first leg.
It was a mixed night for Scottish clubs, as St Johnstone and Aberdeen progressed but Motherwell crashed out to Stjarnan.
St Johnstone needed penalties to see off Luzern after a 1-1 draw at McDiarmid Park, while Aberdeen will now face Real Sociedad following a 2-1 success away at Groningen.
Motherwell could not complete a hat-trick of wins, however, losing out 3-2 after extra-time in Iceland as Stjarnan advanced 5-4 on aggregate.
Hajduk Split survived a late fightback to see off Dundalk 3-2 on aggregate, while Gothenburg progressed despite a 1-0 loss on the night to Gyori thanks to their three-goal advantage from the first leg.
Mlada Boleslav booked a third-round tie with Lyon courtesy of a devastating 4-0 win at Siroki Brijeg and PSV await St Polten, who won 2-0 to beat Botev Plovdiv 3-2 overall.
There were comfortable passages for Shakhter Karagandy, Molde, Zorya, Grodig, Slovan Liberec and Astana but Neftci, Sarajevo and AIK all had to overturn first-leg deficits against Koper, Haugesund and Linfield respectively.
Lech Poznan were another to come from behind and qualify, triumphing 3-0 at home to Nomme Kalju after a 1-0 first-leg reverse, and Spartak Trnava's 3-0 win saw them ease past Zestafoni.
Asteras Tripolis overcame RoPS courtesy of a 4-2 home win, while Petrolul Ploiesti's 3-1 success saw them secure a 5-1 aggregate scoreline against Flamurtari Vlore.
Elsewhere, Esbjerg, Cluj, Diosgyor, Vikingur, Zulte-Waregem, Metalurg Skopje, Rijeka and FH also progressed to the next stage.
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