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."
Shrewsbury 01 Neal 02 Grandison 03 Jacobson 05 Sharps 21 McLaughlin 07 Wright 14 Richards 19 McAllister 26 Sawyers (Taylor - 46' ) 09 Collins (Gornell - 88' ) 10 Morgan Substitutes 25 Smith 06 Cansdell-Sherriff 08 Wroe 23 Taylor 24 Gornell Burton Albion 23 Atkins 03 Webster 04 Stanton (Austin - 63' ) 08 Corbett 33 Ada 13 Taylor Booked 17 Phillips Booked 22 Dyer 24 Harriott (Bolder - 75' Booked ) 09 Richards (Yussuf - 81' ) 10 Zola Substitutes 15 Austin 11 Palmer 21 Gurrieri 26 Bolder 12 Yussuf