Everton had started superbly and deservedly led when Steven Naismith swept in from close range after 12 minutes. But the Toffees' top-four hopes were again undermined as they switched off at a late free-kick and Bassong headed past Tim Howard at the far post in the final minute.
After their frustrating recent results, Everton had appeared to mean business from the outset. Just two minutes had gone when Naismith played in Leon Osman and the recently-capped England midfielder curled an effort narrowly wide of former Everton goalkeeper John Ruddy's post.
Everton's early superiority was rewarded when Naismith emphatically finished off a slick move.
Bryan Oviedo burst down the left after being picked out by a fine ball from Thomas Hitzlsperger and Naismith crashed his low ball into the roof of the net from six yards.
Everton could have doubled their lead as Steven Pienaar, apparently on the right but roving across the field, linked well with Hitzlsperger and Leighton Baines. Baines powered into the box and tested Ruddy with a fierce shot which he could only parry into Osman's path, but the follow-up attempt went over.
Norwich ended the half much better than they started it and had another chance when Bassong won a header at a corner but his effort was well wide.
Everton paid for their missed chances as the Canaries snatched an equaliser just as the fourth official held up his board to indicate four minutes of stoppage time. Javier Garrido curled in a free-kick from the right which dipped over several heads before landing perfectly for Bassong to power a header on target.