The hosts started the match brightly, coming close through Anthony Pilkington after just two minutes.
© PA Photos
The Gunners responded with a chance of their own two minutes later when a fluid move provided Lukas Podolski with the space to shoot, but the Germany international couldn't find the target.
Michael Turner almost extended the Canaries' surprise lead 10 minutes before the break when he made a darting run to the front post to meet a cross, but he couldn't get enough on his header.
Arsenal had a chance to equalise seven minutes into the second half when John Ruddy failed to clear a cross. The ball fell to Olivier Giroud, but the Frenchman was off balance and couldn't direct his shot goalwards.
The Gunners spent the closing stages camped in the Norwich half but couldn't find a way through as Chris Hughton's men held on to claim three valuable points.