Harry Kane struck a late penalty to earn Tottenham a point at Carrow Road and deny Norwich a first Premier League victory in seven matches.
Mario Vrancic opened the scoring with his first Premier League goal for the Canaries before they were denied a second when Teemu Pukki's goal was ruled out by the video assistant referee (VAR).
Spurs improved after the interval and Christian Eriksen drew them level with a stunning free-kick but Norwich retook the lead when Serge Aurier put through his own net.
The visitors had their chances and Dele Alli's neat finish was ruled out for offside before Kane drew Christoph Zimmermann into a lunging challenge inside the area.
The England striker sent Tim Krul the wrong way from the spot to secure a point for the north London side.
Norwich remain bottom of the table and six points from safety, while Spurs are up to fifth, two points behind fourth-placed Chelsea who have a game in hand.
Norwich last won in the Premier League on 23 November, but they started brilliantly against Jose Mourinho's Spurs.
They played with attacking intent from the opening exchanges as Vrancic rolled the exciting Onel Hernandez through on goal, only for Paulo Gazzaniga to sprint off his line to collect the loose ball.
That set the tone and Daniel Farke's side pressed their visitors high up the field. When Juan Foyth hesitated and lost possession deep in his own half, Vrancic carried the ball with urgency before firing past Gazzaniga with a smart finish.
Then came a moment of controversy as Vrancic turned provider to fire a long ball onto the chest of Pukki, who held off Jan Vertonghen to slide the ball into the net, only for the goal to be ruled out.
The Finnish striker appeared to come back onside before the ball was played over the Spurs defence, but VAR adjudged his right arm to be offside.
Spurs hit back after the interval but Farke's men rode the storm and retook the lead as Pukki continued to cause confusion, his effort deflecting from Toby Alderweireld onto Aurier and into the Spurs' net.
As Spurs searched for the equaliser, the hosts demonstrated a steely mentality and dived into tackles - but ultimately did so once too often. Zimmermann lunged into a tired-looking tackle to concede the penalty and Kane punished Norwich.
Points against the so-called 'big six' must be built on if the Canaries are to become the first side bottom of the Premier League at the turn of the year to overcome a six-point deficit from safety and survive.