Robin Quaison had scored two late goals to give Sweden a 3-2 lead in stoppage time but Kamara headed home with virtually the last touch of the game to snatch a point in a pulsating game.
The result leaves Sweden in second place on four points from two games, two points behind leaders Spain who won 2-0 away to Malta, while Norway are fifth on one point.
The Nordic rivals traded thunderous tackles in front of a packed Ullevaal Stadium, with Josh King bringing the home crowd to their feet early on as he hit the bar with a towering header.
King was involved again as Norway took the lead when his shot was blocked into the path of Bjoern Johnsen, who fired home in the 41st minute from what appeared to be an offside position.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
King scored himself in the second half, winning the ball from Albin Ekdal in the centre of the pitch to start a counter-attack before racing into the box to head home Markus Henriksen's cross in the 59th minute.
The Swedes got a lifeline in the 70th when Viktor Claesson was bundled over in the area and then rattled home the rebound after Andreas Granqvist's penalty was saved by Rune Jarstein.
They levelled in the 86th when Quaison's shot took a wicked deflection off defender Haavard Nordtveit and, as the clock ticked past 90 minutes, Quaison then steered Classon's pass beyond Jarstein to put the Swedes ahead.
The drama did not end there and Jarstein made his way forward as Martin Odegaard took a corner deep into stoppage time, with Kamara popping up to nod home a late equaliser for Norway in a thrilling climax to the game.
(Reporting by Philip O'Connor in Stockholm; Editing by Ken Ferris)
