The Magpies went down to 10-men after just nine minutes when Ciaran Clark was shown a red-card for a professional foul on Norwich striker Teemu Pukki.
Despite their personnel disadvantage, Eddie Howe's team went on the front foot and eventually broke the deadlock through a penalty from Callum Wilson.
Toon were desperate for a win but were unable to hold onto their lead when Pukki pounced on a mistake from Martin Dubravka with 11 minutes remaining.
Dubravka flapped at a ball into the box and it eventually fell to Pukki,who unleashed a thunderbolt of a volley past the Slovakian goalkeeper.
Howe's side remain in 20th on the table, three points behind 18th-placed Norwich.
Speaking after the game, Howe was confident his side would win games this season.
"We had a mountain to climb so early in the game. We started brightly 11 v 11. The sending-off has a huge bearing on the match," Howe said.
"Full credit to the players for how they responded. There were some great individual performances.
"I'm so disappointed for everyone connected with the club we couldn't get over the line.
"Defensive resilience [was the best thing]. 11 v 10 for 80 minutes. We defended really well. Norwich made it difficult for us but our midfield coped.
"If we deliver the mentality we showed with 10 men v 11 on a regular basis, we'll win games. No doubt about that.
"The players have had some hard knocks this season and another one tonight with the red card.
"We just have to dust ourselves down, reflect on tonight and then go to Burnley."
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