Oliver Allen opened the scoring for Keith Bertchin's side just 15 minutes into the match at Damson Park and Asa Hall scored a second just past the half hour mark.
The result saw Blues climb to second in the Pontins Holidays Division One Central League table, level on points with leaders Nottingham Forest.
Blues: Doyle, James Blake, Campion, Oji, Painter, Asa Hall, Wright, Kilkenny, Aluko, Allen, Birley
West Brom: Hoult, Watson, Baker, Hodgkiss, Forsyth, R Davies, Chaplow, Elvins, Ellington, Hartson, Nicholson.