Honored and embraced before kickoff by past and present United managers Alex Ferguson and Jose Mourinho, Wenger trudged away from Old Trafford nursing yet another painful loss.
Arsenal conceded an injury-time goal by Marouane Fellaini to lose 2-1 to Manchester United in the Premier League on Sunday.
The winner between Madrid and Arsenal will likely play Marseille, who beat Salzburg 2-0 in their first leg with goals from Florian Thauvin and Clinton N'Jie.
Wenger added that he will not commit to another job while employed by Arsenal, and would not be helping the club in its search for a new manager.
The win was only secured with a flurry of goals in the last 10 minutes, with Aaron Ramsey's strike followed by Alexandre Lacazette's double.
Hours after the 68-year-old coach announced he was stepping down at the end of the season, Gazidis was not giving anything away about the identity of the next occupant of the Arsenal dugout.
Through the thick and thin, Wenger has bared it all. The criticism, the scrutiny, the immense pressure, the Frenchman has seen and achieved it all as an era after 22 years, comes to a glorious end.
Arsene Wenger has won the Premier League title three times and the FA Cup seven times with Arsenal since taking over in October 1996.
Atletico, the 2010 and 2012 winner, will travel to London for the first leg against Arsenal who have never won the Europa League or the UEFA Cup.
This game was the clearest indication yet that Arsene Wenger is now prioritizing the Europa League with victory in the competition securing a route into the Champions League.
Chelsea won't only be surrendering the Premier League trophy to Manchester City but facing a season out of Europe's elite competition.
Marseille progressed after a 2-1 win at Athletic Bilbao that was marred by crowd trouble. Lazio advanced with a 2-0 win at Dinamo Kiev and Leipzig drew 1-1 at Zenit.
Tottenham moved into third, a point above Liverpool, and five points clear of fifth-place Chelsea with eight games remaining in the race for Champions League qualification.
There was another surprise in Germany where two goals from Valon Berisha led Salzburg to a 2-1 victory at Borussia Dortmund.
Chelsea lost in Manchester for the second straight weekend, defeated 1-0 at runaway leader Man City to drop five points adrift of the top four with nine games remaining.
After back-to-back 3-0 losses to Manchester City across two competitions, Arsenal stumbled again following two errors by goalkeeper Petr Cech on the south coast.
Arsenal squandered an early chance to get back into the match in the second half when Pierre-Emerick Aubameyang's penalty was easily saved.
Athletic Bilbao are scheduled to next face Marseille, while Borussia Dortmund will take on Austrian side Salzburg.
Shockingly, Serie A leaders Napoli weren't good enough to advance in the Europa League on Thursday after they were knocked out in Germany by RP Leipzig.
Arsenal powered past Ostersund 3-0, Atletico Madrid thrashed Copenhagen 4-1 and Italian league leader Napoli slipped to a surprise 3-1 home defeat against Leipzig.
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