Villa exited the FA Youth Cup at the third round stage after a bitterly disappointing night at London Road.
Posh made a flying start to this rearranged clash with Jaanai Gordon giving them the lead.
A curling free-kick from Charlie Coulson was prodded home by Gordon after seven minutes.
Soon after, Jevani Brown struck for the home side, who established a 2-0 half-time lead.
His deflected effort went through a crowd of players and found its way into the bottom corner.
In the second period, substitute Levi Wright took advantage of a slip from goalkeeper Craig Hill to prod into an empty net.
And Coulson rounded off the scoring by converting from close range to cap a miserable night for Tony McAndrew's team.
Peterborough will now advance to play Leicester City in round four.
VILLA: Hill, Lyden, Kinsella, Melvin, Bateman, Webb, Graham, Ward, Robinson, Grealish, Calder. Subs: Lewis, Watkins, Crooks, Crowley, Strain.