Albion recorded a second Central League win, coming from behind to claim victory at the Mick George Training Academy.
Rayhaan Tulloch, Chris Willock, Nick Clayton-Phillips and Finn Azaz all netted to secure the points for the Baggies, who claimed a second consecutive 4-3 triumph after beating Mansfield by the same scoreline on their first Central League outing.
Within the first few minutes Tulloch gave Albion the lead, but Frankie Burrows fired the hosts level from distance.
Burrows completed a brace to put the Posh ahead and was brought down in the box to earn Peterborough the chance to go further ahead.
Idris Kanu took advantage from the spot, but Willock halved the hosts’ two-goal advantage just before the break.
Clayton-Phillips hit a long-range strike to fire the Baggies level before Azaz netted to complete the comeback and hand Albion the lead.
The Throstles would see out the victory despite the visitors having the ball in the back of the net once more - only for Rhys Bennett’s effort to be ruled out.
ALBION: House; Meredith, Martinez, Fitzwater, Townsend; Harper, Azaz; Clayton-Phillips, Tulloch, Willock; Soule.
Substitutes: Emery, Harmon, Shotton, Wilding, Asomugha.