Lukaku scored late to add gloss to the scoreline, but United had been far from impressive on the night.
Rashford wastefully blasted high from just inside the area after showing great feet to work his way into the box and, for all their dominance, it took United until the 21st minute to force Scott Carson into a save, the former England goalkeeper flying to his right to keep out Juan Mata's free-kick.
Carson kept out an array of stinging long range efforts, including three from Paul Pogba as United struggled to find a way through a resolute and well-organized visiting defense. Sam Winnall then dragged one wide of the United post just before the half-hour mark as the visitors started to enjoy themselves.
Derby County made a very strong start to the second-half and were clearly growing in confidence; but still living dangerously as Mata, Lukaku and Pogba all had shots saved, blocked and fired narrowly wide respectively five minutes after the restart.
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Lingard, in the form of his life, pulled off a stunning first-time effort from outside the area to break the deadlock on 84 minutes.
Ander Herrera was the next Manchester United player to come close six minutes later but the Spaniard failed to hit target from a set piece from Rashford.
In the second half, Jose Mourinho introduced Romelu Lukaku hooking off Henrikh Mkhitaryan in the process but it was Derby who came close to scoring.
The Red Devils hit the woodwork again minutes later from Romelu Lukaku's header, but a goal wouldn't have counted anyway as the big Belgian was ruled offside.
A second would follow soon following a fast break from the hosts, with Lukaku playing a one-two with Anthony Martial before slotting home.