David Moyes' side had looked good for all three points after Steven Naismith's goal early in the first half gave them the lead.
But Everton never got going in the second half and it was no surprise when Bassong headed home in the final minute.
It was a fifth draw in seven games for Moyes' side but another encouraging result for Norwich after last week's defeat of Manchester United.
Despite being without their leading light Marouane Fellaini, and the likes of Kevin Mirallas and Phil Neville through injury, Everton started brightly and took the lead just 12-minutes in.
New boys Thomas Hitzlsperger and Bryan Oviedo combined well down the left as the Costa Rican broke into the box and squared for Naismith to side-foot home.
Former Toffee John Ruddy had a busy first half producing a string of good saves whilst Tim Howard was a virtual spectator at the other end.
But the second half was a different story as Norwich came out strong and Howard was suddenly asked to keep his side in the game, saving well from Anthony Pilkington before Johnny Heitinga had to clear off the line from Robert Snodgrass.
Everton's only real threat in the second period came from England full-back Leighton Baines, who was a constant thorn in Norwich's side.
Ruddy produced a fine save to deny Baines a wonderful individual effort before the left-back then fired over the bar from a free-kick on the edge of the box.
Everton looked as if they had done enough, but then in the final minute Norwich were level. Baines gave away a free-kick 35 yards out and Javier Garrido whipped it towards the back post where Bassong rose to beat Howard from an acute angle.
Everton remain in fifth with the point, but they miss the chance to close to within one point of Chelsea in fourth. Norwich remain without a win on the road, but Chris Hughton can be more than pleased with his side's display on Merseyside.
|Everton||Team Statistics||Norwich City|
|1||1st Half Goals||0|
|7||Shots on Target||4|
|6||Shots off Target||4|
|61.9||1st Half Poss.||38.1|
|55.4||2nd Half Poss.||44.6|