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Saido Berahino, who scored four times in West Brom's 7-0 third-round win over Gateshead Town, almost broke the deadlock after 10 minutes, but his rising left-footed strike from the edge of the area flew narrowly wide of the near post.
West Brom took the lead in the 25th minute when Berahino slipped the ball through for Anichebe, who snuck the ball inside Darren Randolph's near post from 12 yards.
The forward then doubled his tally as he backed into his marker on the edge of the box before spinning and firing a shot inside the far post.
Anichebe spurned the chance to net a first-half hat-trick with a poor finish inside the area, and Birmingham grabbed a goal back on the stroke of half time when Jonathan Grounds flicked home Lloyd Dyer's left-wing cross.
Myhill soon left the field after a smart save at the feet of Dyer saw him break his finger, and Ben Foster came on the face his old club.
Chris Brunt almost sealed the game for Albion late on, only for his low strike to cannon off the upright.