Albion were three goals to the good and cruising with 41 minutes on the clock thanks to Luke Moore's brace - the first of which came courtesy of a mistake from Posh keeper Joe Lewis - and Chris Brunt`s first of the season.
But Peterborough rallied strongly and Craig Mackail-Smith`s diving header gave the hosts hope at the end of a pulsating opening 45 minutes.
The Championship new-boys flew at the Baggies at the start of the second half and Aaron Mclean pulled another back from close range in the 53rd minute.
London Road erupted shortly after as Posh again had the ball in the net only to see the `goal` rightly ruled out for a foul on Scott Carson.
But Albion closed out the game, despite Graham Dorrans` 90th-minute dismissal, to make it seven points from their opening three games.