Despite Andre Villas-Boas's men enjoying the majority of possession, it was the Canaries that had the better opportunities in the first 45 minutes.
Russell Martin hit the bar with a header from a free kick, before Brad Freidel denied Robert Snodgrass with a double save as the first half came to a close.
The deadlock was broken by Moussa Dembele, who came on to make his debut at the start of the second half.
The Belgian fired home after a smart turn on the corner of the Norwich area.
However, Chris Hughton's men earned a deserved point with just six minutes left when Snodgrass dispatched a half-volley past Freidel.
Tom Huddlestone was sent off late in the game for a 50/50 challenge that the referee had deemed dangerous.