With Olympiakos Piraeus' 2-1 defeat at Paris St Germain, the Greek side and Benfica are equal on seven points apiece in Group C, six behind PSG, and separated by minimal goal difference going into the final group games.
Benfica are at home to PSG while Olympiakos host Anderlecht on December 10.
For the Belgian side, who have taken just one point from five games, all hopes of continuing to play in European tournaments this season evaporated.
Anderlecht started well, taking an early lead in the 18th minute when Chancel Mbemba tucked the ball past Benfica goalkeeper Artur after a fine pass from Fabrice N'Sakala found him in front of goal following a corner.
Scenting a chance at making their mark for the first time in this season's competition, the home side found the net again in the 27th minute but Aleksandar Mitrovic, who had put the ball away, was ruled offside.
Benfica equalised against the run of play in the 34th minute when Nemanja Matic headed in a free kick to protests from Anderlecht players that he was also offside.
Controlling the midfield better for much of the second half, Benfica scored again in the 52nd minute when Mbemba deflected a shot from winger Nicolas Gaitan into his own goal.
With less than 15 minutes to go, Massimo Bruno equalised for the hosts ushering in a frantic finale with chances for both sides to take home the three points.
After Mbemba had a goal disallowed for offside in the 88th, a fast counter by Benfica saw Rodrigo score the winner two minutes later.
"We didn't deserve to lose because we could have made much more of the match. We conceded the goals too easily," Anderlecht's Dutch defender Bram Nuytinck told Belgian broadcaster 2Be.
Anderlecht have had a difficult season so far and are in fifth place in the Belgian league having lost 2-1 at Charleroi at the weekend.
(Editing by Rex Gowar)
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