The lowest-ranked team in the tournament fought valiantly in their meeting with AZ in the Netherlands after falling behind to Stijn Wuytens's opener, and were rewarded late on when Ciaran Kilduff found a leveller.
Zenit St Petersburg battled back from three goals behind in the final 13 minutes to earn all three points in arguably the best tie of the round, as Maccabi Tel Aviv produced an almighty collapse on home soil following the loss of Elazar Dasa.
Dasa was shown red with the scoreline still at 3-1, but Zenit stormed to the most unlikely of 4-3 wins - capped by Luka Djordjevic's goal in the second minute of added time in Israel.
One of the competition's favourites had to settle for a 1-1 draw in their opening tie, with Roma - Italy's fourth-most successful side in terms of major honours - being pegged back early on by Czech champions Viktoria Plzen
There was a shock result at San Siro as fallen giants Inter Milan were stunned 2-0 by Hapoel Be'er Sheva, while elsewhere in Group K Southampton got their campaign off to a winning start with a comfortable 3-0 triumph over Sparta Prague.
England's other Europa participants did not fare as well, though - a much-changed Manchester United falling to a 1-0 defeat against Feyenoord at a reduced capacity De Kuip to make it successive defeats in all competitions under Jose Mourinho.
Europa League results in full: Qarabag FK 2-2 Slovan Liberec, Anderlecht 3-1 Qabala, APOEL 2-1 Astana, Astra 2-3 Austria Vienna, AZ 1-1 Dundalk, Feyenoord 1-0 Man United, Maccabi Tel Aviv 3-4 Zenit, Mainz 05 1-1 St Etienne, Rapid Vienna 3-2 Genk, Sassuolo 3-0 Athletic Bilbao, Viktoria Plzen 1-1 Roma, Young Boys 0-1 Olympiacos, Luhansk 1-1 Fenerbahce, Inter Milan 0-2 Be'er Sheva, Konyaspor 0-1 Shakhtar, Nice 0-1 Schalke, Osmanlispor 2-0 Buceresti, Panathanikos 1-2 Ajax, PAOK 0-0 Fiorentina, Salzburg 0-1 Krasnador, Southampton 3-0 Sparta Prague, Braga 1-1 Gent, Standard Liege 1-1 Celta Vigo, Villarreal 2-1 FC Zurich