But Posh showed what they were made of pulling two back and having two more efforts disallowed.
Roberto Di Matteo's men made the home side pay for some sloppy defending with Luke Moore needing only three minutes to break the deadlock when Posh keeper Joe Lewis allowed a powerful shot to squirm under his body - and worse was to
follow for the hosts.
Wales international defender Craig Morgan was left red-faced after Moore gratefully accepted a simple opportunity to double the Baggies' lead 11 minutes before the break.
Morgan's woeful pass was easily intercepted by Albion front man Chris Wood who released his strike partner to fire past Lewis with minimum fuss.
That goal came after a period of Posh domination in which Craig Mackail-Smith struck an upright either side of two Aaron McLean efforts being saved by Scott Carson during a frantic goalmouth scramble. Mackail-Smith, making his 100th Posh appearance, also had a header ruled out for off-side.
Normal service soon resumed, with Albion striking again four minutes before the break when Chris Brunt tapped home from point-blank range after debutant defender Shaun Cummings had flashed a low delivery across the face of goal.
Any thoughts of an easy ride from then on for Albion were quickly dismissed as Mackail-Smith headed past Carson in first half stoppage time after being picked out by a Shaun Batt cross that begged to be finished off.
Darren Ferguson's men maintained the momentum after the break and pulled back another one with McLean forcing in George Boyd's low 53rd minute cross.
Posh thought they were level when Mackail-Smith bundled the ball into the net only for McLean to be penalised for a foul on Scott Carson.
Morgan slipped again near the end to let in Moore who was denied by Lewis, as was substitute Jerome Thomas after a mazy run.
West Brom midfielder Graham Dorrans was dismissed in stoppage time for a second bookable offence.