Requiring a win for Group D rivals Borussia Dortmund over Galatasaray - which the Bundesliga side duly delivered 4-1 - or a draw at Signal Iduna Park, Arsenal threw away a three-goal lead at the Emirates Stadium.
After a positive Arsenal start, a foul on Danny Welbeck in the penalty area allowed captain Mikel Arteta to open the scoring from the spot 24 minutes in.
And Arsenal were two to the good before the half-hour as Alexis Sanchez lashed past Silvio Proto after his initial free-kick was blocked.
Alex Oxlade-Chamberlain's strong run and finish made it three before the hour mark, but things quickly went downhill for the hosts - Anthony Vanden Borre's double inside 12 second-half minutes setting the wheels in motion.
Aleksandar Mitrovic then had the final say in the 90th minute as Besnik Hasi's side denied Arsenal a place in the knockout stages for a 15th consecutive season - for the time being at least - and kept their own slim hopes of progression alive.
Anderlecht almost opened the scoring after eight minutes when Dennis Praet's effort deflected off Per Mertesacker and into the side-netting, but the hosts responded as the returning Aaron Ramsey - in for Mathieu Flamini – saw a strike blocked and Alexis struck a post before Chancel Mbemba clumsily brought down Welbeck having found himself the wrong side of the Arsenal man.
Skipper Arteta calmly lofted the ball past opposing captain Proto to give Arsene Wenger's side the lead and Alexis made it 2-0 with a fifth goal in his last three games soon after.
The former Barcelona man's long-range strike beat Proto at the near post to leave the visitors in real danger.
Arsenal's dream half ended with a scare as Ramsey sustained a knock to his lower back, but the Welshman emerged apparently unscathed after the interval.
Oxlade-Chamberlain made it three soon after with his first European goal of the season after some slack play from Sacha Kljestan in midfield.
The England international robbed the American and surged past Vanden Borre before slamming the ball beyond Proto, although the experienced full-back then did his best to sour things for Arsenal as he hit back within three minutes from close range.
The 27-year-old was on the end of Andy Kawaya's delivery to pull a goal back for the visitors before reducing the deficit to one from the spot following Nacho Monreal's foul on Mitrovic.
And Mitrovic proved to be Anderlecht's hero, capping a remarkable comeback by heading home Andy Najar's delivery to salvage the Belgians' second point of the group and gain some revenge for Arsenal's last-gasp 2-1 win in Brussels.
