Midfielder Elliott Bennett gave the Canaries the lead on the half hour following good work by Robert Snodgrass before Canada international striker Simeon Jackson doubled the advantage just before the break.
Norwich’s passage through to the next round was confirmed with 20 minutes left when Bennett returned the favour to Snodgrass, who slotted home his fifth goal of the season.
Former Posh players Russell Martin, the Norwich captain, and Ryan Bennett kept their places in a Canaries side showing eight changes, while goalkeeper Declan Rudd was among those given a chance to impress by manager Chris Hughton.
The hosts were weakened by the absence of three key players, with Gaby Zakuani on African Nations Cup duty, in-form striker Dwight Gayle cup-tied and Gavin Tomlin suspended.
They began brightly, though, and Joe Newell may well have scored had Ryan Bennett not deflected his shot just past the post. But Posh’s dominance was short-lived and Snodgrass was the main man as Norwich began to get on top.
In the 30th minute the former Leeds man jinked into the area and set up Elliott Bennett, who poked his shot into the bottom corner.
Posh manager Darren Ferguson made a tactical change, bringing on 17-year-old Jaanai Gordon for Craig Alcock and going to five at the back, but in the 41st minute they conceded again, a long ball from David Fox caught out the home defence and Jackson took his chance expertly with a cool lob over Robert Olejnik.
Ferguson made another change at half-time, sending on Nicky Ajose for Nathaniel Knight-Percival, but the pattern remained the same and Snodgrass was frustrated he could only find the side netting when most of the visiting fans thought he had scored.
There were belated signs of improvement from Posh midway through the half but the game was all over in the 70th minute as Norwich broke away, Elliott Bennett fed Snodgrass and he calmly beat Olejnik.
The midfielder was replaced moments later by Jacob Butterfield, his afternoon’s work done, and, although Posh kept pushing, it was only for a consolation goal.
They almost got it with three minutes to go after Martin had been booked for a foul on Tommy Rowe but Bostwick could only poke McCann’s free-kick into Rudd’s arms – the keeper’s first save of the match.