Glenn Roeder's side survived for 70 minutes with ten men after defender Gary Doherty was dismissed for bringing down Nathan Tyson when the striker was through on goal.
They still took the lead through Matty Pattison in the 23rd minute and, after Paul Anderson had equalised just before the break, a second-half own goal from Luke Chambers gave the Canaries all three points.
Lee Camp had to be on his toes to make an outstanding stop in the second minute, after David Bell had popped up unmarked at the far post to drive a rising shot goalwards from eight yards.
Tyson should have put the home side ahead when Chris Cohen's cross picked him out on the edge of the six-yard box, but the striker headed badly wide.
Camp was at full stretch again at the other end, this time to palm around the post after a deflection sent a Lee Croft cross arrowing towards the near post. But the end-to-end action continued as David Marshall blocked superbly with his legs to thwart Joe Garner from eight yards.
The excitement was still to reach a crescendo however as, in the space of three minutes, Norwich found themselves reduced to ten men only to then take the lead.
Skipper Doherty was shown the red card in the 20th minute after hauling down Tyson as he surged onto the end of a Kelvin Wilson throughball.
Matt Thornhill almost scored from the resulting free-kick, with Marshall palming his low, fiercely struck effort around the post.
But it was Norwich that were to take the lead in the 23rd minute as Wilson's failure to clear gift-wrapped a chance for Pattison, who rammed a volley into the back of the net from ten yards.
Chambers and Garner both came close to a leveller with headers, with Marshall again making smart stops.
But in the 38th minute Forest did make the breakthrough as Garner sent Paul Anderson bursting into the box where he cleverly turned to leave Elliot Omozusi in his wake before bending a right-foot shot beyond the keeper and inside the post.
Half-time substitute Garath McCleary almost had an immediate impact with a deflected shot, while Marshall also had to show quick reactions to make yet another stop from Garner and from Cohen's long-range effort.
But it was Norwich that took the lead for a second time in the 73rd minute as Chambers deflected a Croft cross into the back of his own net at the far post, as Canaries sub Mark Fotheringham pressured.
And the final whistle was greeted with a volley of jeers by the home fans as Forest slipped to the foot of the table, six points from safety.