The 2-0 win lifts Albion four points clear of the relegation zone with three games remaining.
It was also the catalyst for Birmingham manager Gianfranco Zola to resign after just one win in 22 games in charge.
The Aussie attacking midfielder picked up a booking midway through the second half.
Meanwhile, fellow Caltex Socceroo Bailey Wright played a full 90 minutes for Bristol City in their 1-1 draw at Blackburn Rovers.
Wright also earned a yellow card early in the second half.
The point lifts City to 18th in the Championship, four points clear of the relegation zone with three games to play.