Roberto Mancini made eight changes to his starting XI, with a trip to Wembley on Saturday in mind, but he saw City recover from a slow start to stamp their authority on proceedings and take the lead through Edin Dzeko's controlled volley.
The Baggies did all they could to force a leveller, with Markus Rosenberg rattling the crossbar with a fiercely-struck second-half free-kick, but it was not to be for Steve Clarke's side.
Despite the visitors' lack of obvious incentive, coming into the game comfortable in eighth place, they started brightly. Romelu Lukaku had three good chances inside the opening six minutes but failed to trouble Joe Hart, shooting and heading at the keeper and dragging another effort wide.
Carlos Tevez had City's first attempt after linking well with Samir Nasri and cutting inside, but his shot was well off-target.
City settled and Jack Rodwell had a chance after Boaz Myhill failed to deal with an Aleksandar Kolarov free-kick, but shot over from close range.
The hosts then enjoyed a good spell as Rodwell was inches away from turning a Kolarov corner towards goal and Tevez rattled the crossbar with a powerful drive from distance.
The opening goal came moments later as Tevez broke away down the right and clipped back a delicate cross for Dzeko. The Bosnian, guilty of a glaring miss at Swansea on Saturday, watched carefully and guided a well-weighted volley past Myhill.
City maintained the momentum and Rodwell had a shot cleared off the line and another effort deflected wide before Milner drove against the foot of the post.
The break came at a good time for West Brom and they began the second period with renewed vigour.
Rosenborg smashed a free-kick against the bar after being tripped by Javi Garcia on the edge of the area and Hart then saved well from Billy Jones' follow-up header. Hart was in action again soon after, tipping a low effort from James Morrison wide of the post.
West Brom applied pressure for a spell but were unable to break through and City came back at them.
Nasri slipped past three defenders on a run into the box but Tevez could not connect with his cross and Rodwell's attempt was blocked. The ball came back to Rodwell but again he was denied as Myhill saved.
Baggies substitute Shane Long fired into the side-netting with his first touch, Maicon came off the City bench and blazed over, but the game petered out in the closing minutes as Mancini's men held on.