The home side appeared well in control after goals from Sebastien Bassong and Pilkington gave them a two-goal lead only for Craig Gardner to reduce the deficit a minute before the interval.
Martin O'Neill's side were the better in the second period but the Canaries held firm to take their unbeaten run in the Premier League to eight matches.
"I think first half we were brilliant. We passed the ball really well and we've scored two good goals and right at the end of the first half we've conceded a sloppy goal," Pilkington told Sky Sports.
"We didn't think they were in it too much and it's made the second half difficult. But we've shown how strong we are again.
"It's a massive win for us and I think we're eight unbeaten now. We're picking up a lot of points so we're going in the right direction. It's a good place to be at the moment."
Norwich goalkeeper Mark Bunn was critical of himself after he failed to get both hands to the ball for Gardner's goal, although he conceded his overall performance was good.
"I got my fingertips to it but maybe I should have kept it out. That happens in football and I just bounced back in the second half and managed to make a few good saves," he said.
"I was delighted with my performance. It's good to get a couple of saves and I'm getting a run in the team now (in place of the injured John Ruddy) so I want to take my chance."