After an even start the visitors took the lead when Anichebe finished off the inside of the post from Saido Berahino’s pass.
Anichebe made it 2-0 after 35 minutes when he shot into the corner, but Jonathan Grounds reduced the deficit in first-half injury-time when he swept home Lloyd Dyer’s cross.
The Baggies were forced to change their keeper in the second half after an injury to Boaz Myhill but still held on to secure their place in the fifth round.
The win means the Baggies are still unbeaten under Tony Pulis and they began brightly as Berahino brushed the side netting with an angled drive.
They took control against their Championship hosts when Anichebe struck twice in 10 minutes. First, Berahino slipped the striker in and he steered a shot in off the near post on 25 minutes.
The pair combined again 10 minutes later as Berahino found Anichebe in the area and he spun his marker before hitting a shot that deflected into the far corner.
Birmingham had come back from 2-0 down to beat Blyth Spartans 3-2 in the third round and they were given a lifeline in first-half injury-time as debutant Dyer crossed for Grounds to score his first goal for the club.
It lifted the hosts but they needed Darren Randolph to keep it at 2-1 as he produced a superb one-handed save to deny Joleon Lescott three minutes after the re-start.
West Brom were forced into two changes as Myhill was injured in a collision with Dyer and Stephane Sessegnon was hurt after being clattered by Paul Robinson.
Myhill’s replacement Ben Foster tipped substitute Demarai Gray’s rising drive over while at the other end Chris Brunt hit the base of the post from a tight angle.
Birmingham refused to give up and Gray’s rasping 25-yard effort was turned over by Foster but, despite six minutes of added time, they could not find a breakthrough.