Ludogorets Razgrad went through on away goals after a dramatic 2-2 draw at 10-man Partizan Belgrade and Zenit St Petersburg scored three second-half goals to end a brave challenge from AEL Limassol.
Moldova, Azerbaijan and Kazakhstan's hopes of being represented in the group stage for the first time ended as their champions were all eliminated.
Azerbaijan's Qarabag Agdam took a 2-1 lead to Salzburg, only to be sunk by two-first half goals from Martin Hinteregger as they lost 2-0 to the Austrians in the third qualifying round, secondleg tie.
Moldova's Sheriff Tiraspol lost 2-1 on aggregate to Slovan Bratislava after a goalless draw at home to the Slovakians and Kazakhstan's Aktobe were beaten 2-1 at former champions Steaua Bucharest, going out 4-3 on aggregate.
Besiktas, who finished third in the Turkish Super League last season, only qualified in a knock-on effect because champions Fenerbahce are serving the second of a two-year ban over domestic match-fixing in 2011.
Leading 2-1 from the first leg, Besiktas extended their lead after 28 minutes when Mustafa Pektemek raced clear down the left and played in former Hoffenheim, Newcastle and Chelsea forward Ba, who slotted into the net with his right foot.
Elvis Manu lobbed an equaliser for Feyenoord in the 74th minute but Senegalese forward Ba replied with two goals in the last 10 minutes as the Turks won 3-1 to complete a 5-2 aggregate win.
Cypriot league runners-up AEL, who took a 1-0 lead to Russia, had Danielzinho sent off in the 27th minute but held out until 10 minutes into the second half when Venezuelan Jose Rondon headed in for Zenit.
Cypriot champions APOEL Nicosia qualified, however, when Cillian Sheridan and Tomas De Vincenti's penalty gave them a 2-0 win over HJK Helsinki for a 4-2 aggregate win.
Bulgarian champions Ludogorets, under new coach Georgi Dermendzhiev, knocked out Partizan for the second season running.
Brazilian Marcelinho gave them a 2-0 lead with a quickfire brace midway through the first half, only for Partizan to hit back with a Petar Skuletic double in the space of five minutes.
The Serbs lost momentum, however, after their centre back Branislav Trajkovic was shown a red card for slapping an opponent in the face shortly before halftime.
Danish champions Aalborg BK stunned Dinamo Zagreb when Anders Jacosben scored twice, the second a thunderbolt from the edge of the penalty area, to give them a 2-0 away win to overturn the Croatians' 1-0 opening victory.
Other teams to qualify were FC Copenhagen, who beat Dnipro Dnipropetrovsk 2-0 after a goalless draw in the first leg, and 1979 runners-up Malmo, who qualified on away goals after beating Sparta Prague 2-0 for a 4-4 aggregate draw.
- Sports & Recreation
- Partizan Belgrade
- Zenit St Petersburg