A Victor Anichebe double secured progression for Tony Pulis's side, despite Birmingham getting one back at St Andrew's.
Joleon Lescott, meanwhile, took to Twitter after the game to celebrate a tough win.
As the 32-year-old admitted on social media, however, Birmingham made life difficult for West Brom - but the FA Cup is about results not performances, according to the defender.
Lescott's words sound as though they've come out of boss Pulis's mouth! Substance over style and grinding out results over producing performances have been the Welshman's codes of conduct for as long as one can remember.
That is perhaps the biggest reason as to why West Brom have returned to their feet since the 57-year-old took over at the Hawthorns, with the Birmingham win the third out of a possible four under his tenure (although the Gateshead victory in the last round only saw him in the technical area after half-time).
As for Birmingham, the Blues can take hope from their own revival under Gary Rowett - as their display showed they can quite easily mix it with Premier League sides.