Having won the competition five times, the last in 1968, West Brom took a grip of the match early on and, apart from a nervous spell in the middle of the first half, the Championship pace-setters never looked like losing their grip.
Although Posh forced several early corners, West Brom needed only eight minutes to stamp their authority on the game, Roman Bednar claiming his 11th goal of the campaign with an easy finish at the far post following a break crafted on the right by Felipe Teixeira.
Robert Koren gave Tony Mowbray's ambitious outfit a two-goal cushion seven minutes later, forcing the ball over the line from close range after Posh goalkeeper Joe Lewis did well to push a rising shot from James Morrison against a post.
Posh had penalty appeals waved aside after Bostjan Cesar looked to have handled a Craig Mackail-Smith header from George Boyd's cross and veteran Baggies goalkeeper Dean Kiely then tipped over a storming 25-yard effort from Charlie Lee after Dean Keates had tapped through a free-kick.
Posh were then denied by a goal-line clearance when Morrison booted away Craig Morgan's header with Kiely stranded after trying to reach Chris Whelpdale's corner.
Things went from bad to worse for Posh after the break when Morgan was shown a straight red card by referee Jonathan Moss and also gave away a penalty after bringing down Jonathan Greening.
The Albion skipper tried hard to stay on his feet before finally stumbling and it was left to Kevin Phillips, a half-time replacement for Bednar, to step forward and give Lewis no chance with a superbly struck low right-footed effort for his 17th goal of the season.
To their credit League Two promotion contenders Posh continued to try and get players forward, but the service achieved never matched the non-stop running of Mackail-Smith or the equally hard graft of Whelpdale.