Keith Bertschin's side were 2-0 ahead inside the first 20 minutes at The Hawthorns courtesy of a brace of goals from Finnish international striker Mikael Forssell.
Steffan Morrison pulled a goal back for the hosts on 25 minutes but Blues restored their two-goal advantage through Neil Danns six minutes before the half time interval.
Substitute Kames McPike scored Blues fourth goal just two minutes before the final whistle.
Blues skipper Damien Johnson, who has been sidelined with a hamstring injury for the past three weeks, completed just over sixty minutes of the game whilst Forssell lasted the whole match.
West Brom Res: Daniels, Baker, Forsyth, Manchester, Wallwork, Worrall, Sissoko, Morrison, Hartson, Compton. Subs: Bateman, D. Nicholson, Mason, Nardiello, McGovern.
Blues Res: Maik Taylor, Johnson, Campion, Pearce, Tebily, Kilkenny , Danns, Nafti, Campbell, Forssell, Gray. Subs: Legzins, Dicker, McPike, Wilson Howland.