Dale had an uphill task from the seventh minute when referee Seb Stockbridge showed visiting skipper Peter Cavanagh a straight red for a high challenge on Robbie Weir - only minutes after a challenge that had left the Burton midfielder needing treatment for a facial injury.
The hosts took time to settle but finally made the breakthrough in the 44th minute when Lancashire slid the ball into his own net as he attempted to intercept a Billy Kee pass in the box.
The Brewers had chances to increase their lead early in the second half but Kee fired narrowly wide and Chris Hussey failed to convert a headed chance.
Dale kept fighting, however, and tried to take the game to their opponents despite their numerical disadvantage.
Indeed, George Donnelly went close to grabbing the equaliser with a fierce 20-yard shot which went over the top.
Defender Ian Sharps could have made it more comfortable for Burton, though, but he headed inches too high in the 85th minute before substitute Michael Symes somehow fired over in front an open goal.