Lukaku scored late to add gloss to the scoreline, but United had been far from impressive on the night.
Three minutes later, Pogba had a chance to send United ahead but his curling shot from just outside the box towards the top corner missed wide.
Lingard and Paul Pogba both had early attempts for a United team that started on the front foot at Old Trafford.
Lingard and Lukaku spared the home side's blushes, but questions will remain over United's ability to get over the line in games, with the 2-0 win over Everton last time out ending a run of four matches without a victory - including a shock 2-1 defeat to Bristol City in the EFL Cup quarter-finals.
Romelu Lukaku, who had replaced Mkhitaryan at the break, then had his goal-bound shot blocked as Pogba curled off target in a good spell of attacking intent from United.
In the 24th minute, United earned a free-kick on the edge of the Derby box but Mata's effort was saved brilliantly by Carson.
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.
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.
United also got good news regarding Romelu Lukaku, as the Belgian worldwide will be fit and available to play after suffering a scary blow to the head against Southampton on December 30.
Having set up the first, Lukaku got on the scoresheet himself by finishing off a quick counter-attack after a one-two with Anthony Martial to ensure United avoided a troublesome replay at the end of a hectic festive schedule.