Alan Irvine's side continued their 100 per cent start to their season preparations after beating Bury on Saturday, with Sebastien Pocognoli making his debut for the club
Having beaten Bury 2-0 behind closed doors on Saturday, Alan Irvine's side beat another League Two opponent thanks to first-half efforts from Anichebe and Mulumbu.
Scott Vernon gave the Premier League side a scare when he cancelled out Anichebe's opener, but West Brom's strength in depth and quality ultimately proved the difference at New Meadow.
Belgium international defender Sebastien Pocognoli made his West Brom debut following a move from Hannover at the weekend, while a tight calf kept Joleon Lescott, another of the club's close-season signings, sidelined.
Albion made a bright start and were ahead after 12 minutes as Anichebe slotted home Stephane Sessegnon's cutback from eight yards.
Vernon, though, hauled Shrewsbury level on the half-hour mark when he powered past Boaz Myhill from close range after getting on the end of Ashley Vincent's pass.
However, parity lasted just four minutes as Mulumbu put the visitors back in front. Sessegnon was again heavily involved in the build-up, this time diverting Anichebe's cross into the path of Mulumbu, who beat goalkeeper Mark Halstead.
Both managers made wholesale changes at the break and Shrewsbury substitute goalkeeper Callum Burton reacted well to keep out a Saido Berahino effort 16 minutes from time in the only real chance of note in the second period.