The Rams started brightly and took the lead in the 11th minute when George Thorne won the ball in the middle of the park and threaded a pass through for Darren Bent, who fired his effort into the net.
Bent almost notched his second goal of the game when Cyrus Christie knocked the ball into his path down the right and the striker drilled a low strike at the net, which was well blocked.
Abdenasser El Khayati almost got a goal back for Burton when he drilled a volley at goal from distance, but Scott Carson made the save.
The Rams pressed forward again and came close to adding to the scoreline through Tom Ince and Weimann, but both players were denied by Remi Matthews.
Jerome Binnom-Williams almost found an equaliser when he raided through into the box, but he lost his composure at the vital moment, blazing his effort over the bar from 12 yards.
Derby secured their victory as Russell latched on to a shot at goal from Jeff Hendrick and placed his effort past Matthews.