The Turkish club — which is still threatened with expulsion from the competition by UEFA in a match-fixing case — won 3-1 in Istanbul after trailing early, and progressed 4-2 on aggregate.
Lyon, another Champions League regular, endured a nervy evening against Grasshoppers Zurich, until highly rated prospect Clement Grenier's 82nd-minute goal secured a 2-0 aggregate win.
Basel maintained Swiss interest in the competition, despite letting a three-goal lead slip at Maccabi Tel Aviv. A 3-3 draw meant Basel advanced 4-3 on aggregate.
Also advancing to Friday's playoffs draw: Steaua Bucharest, the 1986 European champion, Maribor of Slovenia and Ludogorets of Bulgaria.
Shakhter Karagandy of Kazakhstan joined them, and the No. 324 in UEFA's club rankings will likely be the lowliest team left after the third qualifying round is completed Wednesday.
Fenerbahce started as favourite at home, after a stoppage-time equalizer earned a 1-1 draw in Austria last week, but gave back that away goal when Salzburg's Jonathan Soriano scored in the fourth minute.
Portugal midfielder Raul Meireles began the comeback in the eight minute, Moussa Sow put his team ahead in the 17th and Pierre Webo put Fenerbahce in full control in the 34th.
Fenerbahce continues in the competition while awaiting a hearing date at the Court of Arbitration for Sport to appeal a two-year ban from UEFA competitions. It was imposed because club officials were convicted in a criminal trial of helping fix Turkish league matches in 2011.
Basel expected a hostile night in Tel-Aviv, where host Maccabi had warned its fans to expect UEFA disciplinary action if the visitors' two Egyptian players were verbally abused.
Still, the Swiss side enjoyed a commanding start. Fabian Schaer scored a fifth-minute penalty, then Egypt midfielder Mohamad Salah scored in the 21st with a neat finish. He kissed the Bloomfield Stadium turf in celebration.
Basel's Chilean midfielder Marcelo Diaz scored with a sweet, curling shot in the 32nd to build a seemingly unassailable four-goal overall advantage.
But Maccabi pulled back two goals within five minutes, through Schaer's own goal and Eran Zahavi's jinking solo run. The match was levelled in the 54th by forward Maharan Radi but Basel goalkeeper Yann Sommer helped repel a wave of late attacks.
With home wins rare, Romanian champion Steaua needed only a 1-1 draw against Dinamo Tbilisi to advance 3-1 on aggregate.
Maribor's 0-0 draw with APOEL, after a 1-1 draw in Cyprus, was enough to advance on the away goals rule. Ludogorets beat Serbian champion Partizan 1-0 in Belgrade to complete a 3-1 aggregate victory. Hristo Zlatinski scored an 88th-minute penalty.
Shakhter lost 3-2 in Albania against Skenderbeu, but went through 5-3 overall.
Basel, Maribor, Ludogorets, Shakhter and Steaua enter Friday's draw in the section reserved for national champions.
Fenerbahce joins Lyon in the section for non-champions. Potential opponents are AC Milan, Arsenal, Real Sociedad, Schalke and Portuguese newcomer Pacos de Ferreira which enter the competition in the playoff round.
The 10 playoff winners join 22 elite teams already qualified for the lucrative group stage. That draw is made Aug. 29 in Monaco.Suggest a correction