The defeat in the League Two encounter at the Pirelli Stadium put an end to Rovers' five-game unbeaten run.
The Pirates were always going to be up against it after being reduced to ten men after just 36 minutes when striker Chris Beardsley was dismissed.
Burton took advantage of their numerical superiority and went ahead just before the hour-mark.
Rovers eventually ended the game with just nine players on the pitch when John-Joe O'Toole saw red in stoppage time.
John Ward's side remain in 22nd place in League Two, a point off the drop zone.
Elsewhere, Amari'i Bell played the full 90 minutes for Nuneaton Town, as they beat Chester 1-0 on home soil.
The young full-back's efforts saw him win the man-of-the-match award.
A late Louis Moult goal was enough to overcome the Conference Premier.
Nuneaton are just a point outside the Skrill Premier League play-off zone.