Skenderbeu also qualified, putting themselves within two matches of becoming the first Albanian side to reach the group stage, after beating Milsami in another of the third round, second leg ties.
Former European champions Celtic went through after grinding out a 0-0 draw against Qarabag in the sweltering heat of Azerbaijan.
CSKA hit back from two goals behind at Sparta Prague to win 3-2 and qualify 5-4 on aggregate against the demoralised Czechs, who had only needed a goalless draw to progress.
Russian international Alan Dzagoev scored the winner in the 76th minute after two goals from Ahmed Musa had hauled the Russians back into the match.
Astana, having drawn 0-0 in the first leg away to Helsinki, also fell two goals behind at home but hit back to lead 3-2 before Patrick Twumasi, one of their goalscorers, was sent off in the 70th minute.
Erfan Zeneli converted an 86th-minute penalty to level at 3-3 and put the Finns ahead on away goals, only for Evgeniy Postnikov to score a stoppage-time winner for the Kazakh champions.
Serbian champions Partizan went through after a roller-coaster 4-2 home win over 10-man Steaua Bucharest gave them a 5-3 aggregate win over the 1986 European Cup winners.
Trailing 2-1 at halftime, Partizan missed a penalty when Valeri Bojinov hit the post.
But Steaua fell apart after captain Fernando Varela was sent off and Partizan hit back with three goals in the final half hour through Marko Jevtovic, Andrija Zivkovic and substitute Nikola Trujic
Swedish champions Malmo needed only 14 minutes to cancel out a two-goal first leg deficit at home to Salzburg as goals from Nikola Djurdjic and Markus Rosenberg had the Austrian champions reeling.
Although the visitors took control, Vladimir Rodic pounced on a defensive blunder to score the third three minutes before halftime to give Malmo a 3-0 win and 3-2 on aggregate.
It was the eighth time in a row that Salzburg have gone out in the qualifying rounds since Red Bull took over the club in 2005.
Celtic went through 1-0 on aggregate while Skenderbeu's 2-0 win over the Moldovan champions gave them a 4-0 aggregate win.
Shakhtar Donetsk brushed aside Fenerbahce 3-0 to qualify and Brugge beat Panathinaikos by the same score to win 4-2 on aggregate.
BATE Borisov, who beat Videoton 1-0 for a 2-1 aggregate win, Maccabi Tel Aviv and FC Basel also qualified.
(Additional reporting by Zoran Milosavljevic; editing by Toby Davis)
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