Defender Kelvin Langmead broke the deadlock when rifling home from close range in the 24th minute after Ryan Bennett had nodded a half-cleared corner into his path.
Langmead and striker Craig Mackail-Smith were both denied at point-blank range by County keeper Matt Glennon before the break, but Posh quickly seized the initiative following the restart.
Mackail-Smith coolly beat Glennon in a one-on-one situation after being released by George Boyd in the 48th minute and wideman Lee Tomlin blasted a third from 20 yards eight minutes later.
Substitute Greg Tansey cracked a deserved County consolation when his long-range drive flew in off the inside of a post in the 72 minute and the same player saw a header brilliantly parried onto the bar by Posh keeper Joe Lewis moments later.
But the hosts sealed a trip to Bury in the next round when hard-working hitman Aaron Mclean sent a low drive fizzing past Glennon from the edge of the box in stoppage time, although the victory was marred by a serious injury suffered by Dave Hibbert.