Steven Naismith powered home an early effort for David Moyes' men as the home team looked set for victory, but Chris Hughton's side struck with a minute to go to rescue a point
The Toffees struck after 11 minutes following good work from Bryan Oviedo on the left wing, with the Canaries' right-back Steven Whittaker unable to track the Costa Rican as he fed the Scotland midfielder for a simple finish.
However, the visitors kept pushing for an equaliser in the second half, and were rewarded as Bassong powered in a header off the bar with a minute to go following a controversial free kick.
The Toffees were looking to bounce back from a disappointing 2-1 defeat away to Reading, and while Hughton's men were riding high on confidence after going six games unbeaten, it was Everton who started the stronger of the two.
Leon Osman nearly opened the scoring after 70 seconds, sending a lovely left-footed curling effort just wide of John Ruddy's right-hand post.
Oviedo was handed a start on the left and he rewarded Moyes' faith by setting up Naismith for the opener. A high through-ball saw Whittaker badly caught out, and the Costa Rica winger pulled the ball back for the Scotland international to slam home past Paul Ruddy from seven yards.
Leighton Baines provided a constant threat down the left for the hosts, and Steven Pienaar teed up the left-back for a low drive which brought a fine one-handed stop from Ruddy, with Osman only able to volley the rebound over.
Controversy followed soon after as Naismith went down in the Norwich penalty area under a challenge from Alex Tettey. Referee Mike Jones waved play-on, and replays showed the midfielder won the ball cleanly.
Norwich could, however, have been level after 24 minutes - Grant Holt surged past three challenges, but the English forward could only shank his left-footed effort wide of Tim Howard's right-hand post with the goal at his mercy.
Sebastien Bassong wasted a great chance to level after 40 minutes, heading an in-swinging corner wide from six yards - but the fact Howard was left without a save to make in the half revealed the lack of invention from the visitors.
Norwich started the second half confidently, and Anthony Pilkington tested Howard for the first time with a curling 25-yard free kick forcing the American into a scrambling block.
John Heitinga denied Snodgrass an equaliser by brilliantly clearing the winger's goal-bound finish off the line after 68 minutes.
Baines continued his exceptional form by driving forward and bringing another good stop from Ruddy from 20 yards, and the goalkeeper was forced off with an injury with Mark Bunn playing the last 10 minutes.
Pilkington brought a fine stop from Howard with six minutes remaining, but Bassong snatched a point for Norwich by forcing a header home after 89 minutes - although Moyes will surely dispute the nature of the free kick which led to the late leveller.