Mark Cooper’s men frustrated Albion for an hour but a Luke Moore strike and an own goal from Posh centre-half Ryan Bennett ensured that Roberto Di Matteo’s side moved back into second place.
Newcastle, Nottingham Forest and Cardiff all failed to win in early kick-offs on Boxing Day, leaving Albion eight points behind the Magpies.
In the 61st minute, Graham Dorrans played in Moore with a clever reverse pass and he took a touch before sending in a left-footed shot which deflected past the dive of Joe Lewis.
Bennett handed Albion a second in the 67th minute. Dorrans fired a shot into the turf and Bennett – under pressure from Simon Cox – stretched to send a header past Lewis and into his own net.