Ole Gunnar Solskjaer made six changes in Sergio Romero, Eric Baily, Diogo Dalot, Juan Mata, Jesse Lingard and Odion Ighalo, while Wayne Rooney featured against his former side.
United were off to a quiet start, as Rooney forced Romero into the game's first save with a free-kick from the far side following the opening 20 minutes.
Ighalo provided United's first threat of the evening when he got behind defence to bring Ross Atkins to a smart save following a close-range effort after the half-hour mark.
United evidently had their noses in front after Luke Shaw volleyed in off the ground and into the far corner on 33 minutes after the ball bobbled towards the near side area.
Ighalo put United further ahead as he held possession in the area from Shaw's defence-splitting pass to find the bottom corner five minutes to halftime.
Derby were forced into a change on 43 minutes, as Tom Lawrence signalled removal following a knock and was replaced by Morgan Whittaker.
Romero was put into action minutes into second half as he got down to smother Max Bird's low deflected shot, whilst Martyn Waghorn missed the far corner with a pin-point header.
Shaw looked for his second with a low shot on 53 minutes after sneaking ahead of the defence, but his effort failed to get past the boot of Atkins.
The Red Devils grabbed a third after 70 minutes when Ighalo showed sharp instincts to riffle home a rebound inside the six-yard area.
Derby pushed for a consolation afterwards, but Romero tipped Rooney's free-kick for a corner in added time as United set up a quarter-final date with Norwich City.