It also took them eight points clear of United, whose second successive loss – after managing back-to-back wins in mid-October – has them languishing third-from-bottom after 13 games.
Morris compounded what has been a miserable start to the Blades’ season breaking the first-half deadlock with a glancing header from a Joe Ward corner.
Ben Brereton Diaz went close to doubling the visitors’ advantage against the side he featured on loan for last season, but the Chile international was denied by goalkeeper Cooper.
A fine tackle from Matt Clarke on Callum O’Hare inside the box just before the break ensured the Rams held a slender half-time lead.
An aimless backpass from Sydie Peck gifted Morris the chance to pickpocket possession to score his second of the afternoon a minute after the restart.
The former Luton Town, Norwich City and Barnsley forward then made it a milestone afternoon with a successful spot-kick to complete the first hat-trick haul of a career that has spanned 11 years and 10 different clubs.
He would go on to be substituted off to wild cheers from the Derby fans late on, but not before O’Hare had grabbed a consolation goal for the hosts after meeting a long throw into the box before wriggling into space to score.