Albion made the best of starts with a goal after just eight minutes. Justin Richards skinned Rotherham for pace, slipped a pass wide to Cleveland Taylor and Richards continued his run to turn in Taylor's drive from close range.
However it looked for all the world that the forward was in an offside position but the goal was given.
And the visitors could have had a second after 20 minutes but Billy Kee's goal-bound shot from Taylor's corner was deflected wide.
Rotherham struggled going forward before half time, but a clever short corner after 21 minutes by Danny Schofield nearly produced a goal when Ben Pringle slipped the ball back to him and his shot was touched over the bar by Ross Atkins.
But the home side's best chance came after 34 minutes when Pringle headed down Schofield's cross only for Lewis Grabban to sky his shot high over the bar.
Grabban almost made amends four minutes after the re-start but his left footed shot from distance hit the bar and went over.
And his effort sparked a Rotherham revival. Alex Revell's header was scrambled clear by Atkins and the keeper followed that up with two magnificent saves to deny substitutes Marcus Marshall and Gareth Evans who looked a danger on either flank.
But some resolute defending by the Brewers, who readily pulled everyone back behidn the ball and settled for their only goal of the afternoon, saw them through.
Source: DSG