Hope Apkan gave the Brewers the lead when he nodded the ball past Eldin Jakupovic after 17 minutes.
Just 12 minutes later, Lucas Atkins doubled the lead for the Championship minnows.
Mahrez, who has been the subject of two failed bids from Roma, was an energetic presence before being replaced by Demerai Gray after sixty minutes.
Not long after, Harry Maguire blasted home his debut goal for the club. However, the Foxes failed to find an equaliser before the end of the match.
Craig Shakespeare's side take on Borussia Monchengladbach at the King Power on Friday.