Lukaku scored late to add gloss to the scoreline, but United had been far from impressive on the night.
Manchester United take on Derby in the FA Cup third round at Old Trafford tomorrow night, when shot-stopper David De Gea will be handed a rest. That's 11 goals this season for Lingard, and another beauty. But Sam Winnall failed to profit as he dragged his shot wide.
The Red Devils had more options up front, but their best chances were still falling to Rashford, who squandered a pair of opportunities after being put through by Pogba and Lingard, respectively.
Tom Lawrence almost sent Derby ahead two minutes before the hour with a thumping shot from just outside the box but Sergio Romero made a brilliant save at the bottom corner.
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.
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.
Mourinho has lived in the Lowry Hotel in Salford since taking charge of Manchester United in 2016
Manchester United had the first attempt on goal through captain Pogba firing a straight shot at Carson in the second minute.
Rashford thought he had found the breakthrough in the 37th minute with a header from the centre of the box but goalkeeper Scott Carson was equal to the task and kept his side in the match.
At the other end, Marcus Olsson's clever flicked header from Johnny Russell's left-wing cross forced Sergio Romero into his first action of the evening.
The barrage continued and Pogba wastefully side-footed wide with the goal gaping when following up Lingard's saved effort, before Derby finally yielded with six minutes to play.
A second would follow soon following a fast break from the hosts, with Lukaku playing a one-two with Anthony Martial before slotting home.