The Canaries went into the game on a crest of a wave after a fine run that included a derby win at Ipswich and they looked good for another three points when Lewis Grabban scored against his former club after just 14 minutes.
But Eddie Howe’s side retained their composure and levelled just before half-time through Callum Wilson, the main who replaced Grabban at Dean Court.
The second half was a tight-fought affair with clear-cut chances at a premium and in the end a point apiece was a more than fair outcome.
Bournemouth quickly settled to their task in what was their first league visit to Carrow Road for over 54 years and had the first effort of the game, a long-range drive from Dan Gosling that flew well over.
The Canaries, by contrast, started slowly and when they opened the scoring on 14 minutes it was from their first opportunity.
Nathan Redmond notched an impressive assist with an inch-perfect cross from the right and Grabban simply had to guide the ball in with his head from the edge of the six-yard box, after finding a pocket of space.
It was a goal that flattered the Canaries on the early balance of play, with the hosts some way below their best – although Kyle Lafferty did force Lee Camp into a save on 27 minutes with a crisply struck first-time shot.
Norwich continued to look the more dangerous side as the game progressed, with Lafferty firing just over after neat inter-play in the box and Camp palming away a Redmond free-kick. But it was the visitors who stunned the home fans by equalising on the stroke of half-time.
Having passed the ball around nicely without ever posing a threat, the Cherries suddenly round a cutting edge as Junior Stanislaus picked out the overlapping Simon Francis, who crossed low for Wilson to convert at the back post.
The pattern of neat passing and only the occasion goalmouth incident continued after the break with another big crowd having to wait until the hour mark for the next effort on target, a snapshot from Dan Gosling that was easily gathered by City keeper John Ruddy.
It was Norwich who were having more of the ball however, with substitutes Josh Murphy and Cameron Jerome freshening things up and the latter seeing a header deflected just wide.
At the other end Cherries’ substitute Mark Pugh fired straight at Ruddy after a quick break, but the Canaries kept plugging away and nearly grabbed a winner two minutes from time when Bradley Johnson’s 28-yard free-kick forced a good save out of Camp.