Ian Sharps netted the only goal as Shrewsbury strengthened their hopes of a top-three finish and consigned Burton to their 13th game without a win.
Romaine Sawyers and Marvin Morgan both had chances to break the deadlock in a quiet first half.
Defender Sharps notched his second goal of the season shortly after the break when he headed in Mark Wright's inviting corner.
And James Collins saw his effort denied by Andrew Corbett's goalline clearance.
Shrewsbury, who are a point clear of fourth-placed Cheltenham, have now equalled their club record of going 26 home games unbeaten in all competitions.
VIEWS FROM THE DRESSING ROOM
Burton manager Paul Peschisolido told BBC Radio Derby:
"We put in a rugged performance and we had to defend well at times. I felt the commitment was there and I can't be disheartened by that.
"We didn't cause them that many problems, nor has anyone in the last 26 games they've had here.
"So there's no disgrace in losing to this kind of quality, especially on their own home ground.
"I would've liked to have seen us put up a little bit more attacking, but the commitment was there, so I'm not going to be too disheartened."