Norwich started the better of the two sides and attacked in the first minute, although Anthony Pilkington's shot was blocked well by the Stoke defence.
The Canaries continued to push forward and came close again on nine minutes through Ryan Bennett, who saw his effort saved.
Jonny Howson had an effort for the away side on 25 minutes, but the Potters' defence were alert to the danger and closed down the midfielder.
After 33 minutes, Marko Arnautovic found some room for a shot, but he could not find the target with his effort.
Norwich took the lead a minute later when Pilkington teed up Howson, who fired past Asmir Begovic from 25 yards to make it 1-0 to Chris Hughton's men.