The Belgium international flicked a header past Gunners goalkeeper David Ospina from an Ashley Young cross.
Just three minutes earlier he had headed against the post, with Marcus Rashford bundling home the rebound from an offside position.
United had opened the scoring on 16 minutes when a header from former Arsenal player Alexis Sanchez was knocked on to the post by Hector Bellerin and Paul Pogba stabbed home the rebound for his third goal in four league games.
The hosts lost striker Romelu Lukaku to injury early in the second half, and moments later Henrikh Mkhitaryan was left in plenty of space to weave and strike through the legs of Victor Lindelof.
The game’s tempo petered out after the equaliser – until Fellaini popped up to snatch the win for United, and secure Champions League football for next season.