The hosts began slowly, but they soon had control of the match and should have taken the lead through Wes Hoolahan, while Gabriel Obertan forced Mark Bunn into a good save.
A similar pattern emerged after the break, but this time Newcastle created even less in attack and needed Tim Krul and the post to prevent goals from Anthony Pilkington and Russell Martin, respectively.
The two sides made a raft of changes following their FA Cup ties last weekend as the likes of Sebastien Bassong, Javier Garrido, Pilkington and Wes Hoolahan all returned to the Norwich team, while Simeon Jackson retained his place for a rare Premier League start.
Newcastle welcomed back first-team regulars such as Papiss Cisse, Fabricio Coloccini and Jonas Gutierrez, while the Ameobi brothers – Shola and Sammy – and James Tavernier dropped out. New signing Mathieu Debuchy also started, making his debut following a £6 million move from Lille.
The visitors almost created the first chance of the match in the opening 10 seconds as Vurnon Anita smashed the ball into the Norwich area from his own half and Bassong made the bizarre decision to leave it, allowing Gabriel Obertan the chance to cross from a great position, but his effort was awful and Mark Bunn collected.
After that slow start, Norwich began to take control of the match in midfield, stringing together some lovely moves, while Pilkington and Hoolahan were terrorising Debuchy, but it was Newcastle who went close next as Obertan spun and shot on the edge of the area, only for Bunn to palm it away.
Despite having the better of the first half, Norwich's problem seemed to be that they had too many creators and not enough finishers. Soon a counterattack gave them their best opportunity as Hoolahan surged forward before playing in Jackson to his right, before having the return ball poked behind by James Perch as he shaped to shoot.
There was still time for Norwich to get an effort on target before half-time as a Pilkington cross was met by Snodgrass’ head, but Jackson couldn’t quite get a touch on it and Tim Krul collected with ease.
The second-half followed much the same pattern as the first period, and Norwich continued to look considerably more dangerous in attack, but still they could not capitalise.
Soon, however, Snodgrass gambled on a long ball from Russell Martin and tried to spin around Coloccini, but the Argentine got a toe in, poking the ball back to Pilkington who fired his shot hopelessly over.
Martin soon had an effort of his own to shout about as he brought the ball under control about 30-yards out after a bizarre period of play that resembled head tennis, and then lashed a half-volley against Krul’s post with the goalkeeper flapping for his life.
It was not long before Krul was forced into action. Snodgrass cut in from the right and picked out Pilkington on the left, who beat Debuchy with an excellent turn before shooting, but there was Krul to make the save and the visitors were able to hang on to save a point.