Grady Diangana headed the Baggies in front five minutes after the break before Matt Clarke doubled the advantage following a scramble in the box.
Karlan Grant made it 3-0 when he stroked home from the penalty spot, after Callum Robinson had been brought down.
Rayhaan Tulloch then completed the rout, firing high into the net after a bursting run from fellow substitute Kenneth Zohore.
Albion were missing Matt Phillips, with the winger self-isolating, while there was once again now Matheus Pereira in the visiting squad, the playmaker likely to leave before the end of the transfer window.
Blues made the brighter start and good defending was required from Kyle Bartley to clear Ivan Sanchez’s dangerous cross, after Clarke had been robbed on the left wing.
Gradually Albion began to find some rhythm. Robinson came close to finding Diangana with a dinked ball over the top, before Grant headed narrowly off target from Darnell Furlong’s cross.
At the other end, Clarke was alert to head Jeremie Bela’s teasing cross over his own bar with Lukas Jutkiewicz waiting to pounce.
As half-time approached, Grant drew the first serious save from either goalkeeper when he cut in from the left and fired on goal, Wolves loanee Matija Sarkic tipping the shot round the post.
Blues had an escape early in the second half when Sarkic was unable to hold Furlong’s corner, Robinson unable to turn the loose ball home as his shot was blocked.
Five minutes after the break Albion were ahead. Furlong crossed left-footed from the right and Diangana nipped in front of Sarkic to guide his header over the keeper and inside the far post.
Sarkic then prevented Grant from quickly doubling the lead when he saved a low shot with his legs following a rapid Albion counter.
But the keeper’s day took another turn for the worse on 69 minutes when he was at fault for Albion’s second. Sarkic came for but got nowhere near Alex Mowatt’s inswinging free-kick and after a scramble where Robinson was close to stabbing home, Clarke eventually nodded in from close range.
The Baggies were well in the ascendancy and soon led by three. Robinson went through on goal and after he tumbled under the challenge of Roberts, referee Stephen Martin pointed to the spot. Grant stepped up to send Sarkic the wrong way and prompt more celebrations among the 1,887 travelling supporters.
Blues substitute George Friend squandered a great chance to score when he headed off target. Moments later the ball was in the net at the other end, Tulloch putting the final touch on a fine move when he converted Zohore’s pull-back.
Blues (4-2-3-1): Sarkic, Colin, Roberts (Friend 85), Dean, Pedersen, Woods, Gardner, Bela, Sanchez (Chong 66), Leko (Hogan 66), Jutkiewicz (Aneke 71) Subs not used: Friend, Lakin, Walker, Oakley, Trueman (gk).
Albion (3-4-3): Button, O'Shea (Ajayi 76), Bartley, Clarke (Kipre 76), Furlong, Livermore (Sawyers 82), Mowatt, Townsend, Robinson (Zohore 76), Grant, Diangana (Tulloch 82) Subs not used: Taylor, Ingram, Ashworth, Cann (gk).