The Blues' stand-in stopper kept out Lee Angol's shot from close range with the score still all square, which proved a pivotal moment in the match as the hosts soon ran away with it.
Despite their efforts, though, third-tier Peterborough were unable to muster a comeback and conceded a fourth when Pedro doubled his tally for the day from the edge of the box.
Speaking after the match, McCann said: "I thought we were excellent. I thought we had two or three really good chances. I couldn't believe Begovic saved from Angol.
"The biggest difference is how clinical they are. It is something we need to work on. I thought the boys were outstanding and in the second half we matched them."
Peterborough, currently ninth in League One, return to action next weekend with a trip to face Bury.