England midfielder Lingard let fly from outside the area with six minutes remaining for his fifth goal in his past six outings to finally break the resistance of the spirited Rams.
It was a huge relief for a United side that appeared destined for a frustrating third-round replay having wasted a glut of chances, and Lukaku made sure of the win in the closing stages.
Marcus Rashford was the guiltiest party in the first half with a poor point-blank header that he tamely nodded against the post.
The England forward was unlucky to see another effort crash against the woodwork in the second half, while Paul Pogba was also profligate.
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.
Pogba dominated the midfield in the early stages and he teed up Lingard to fire over the top from 10 yards in just the third minute.
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.
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.
Derby briefly took United out of their rhythm, but Rashford missed a gilt-edged chance when he nodded Henrikh Mkhitaryan's pin-point cross against the right post from just outside the six-yard box.
Carson was brilliantly equal to a long-range Pogba free-kick just before the break, and shortly after the restart Luke Shaw's head deflected Tom Huddlestone's fierce drive away from United's goal.
Rashford thought he had finally broken the deadlock only for his effort across the face of goal from the right to cannon off the far post, while Lukaku – returning from a head injury sustained against Southampton last Saturday – also struck the woodwork, but was flagged offside in any event.
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.
Lukaku played the ball into Lingard's path, and the attacking midfielder hit an unstoppable shot into the top-right corner.
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.
Key Opta stats:
- Manchester United have now progressed from 31 of their last 33 FA Cup third round ties.
- Jose Mourinho has won 12 of his 13 matches against lower league opposition in the FA Cup, failing only against Bradford City in January 2015.
- Jesse Lingard has scored 11 goals in all competitions this season – as many as he scored in 2015-16 and 2016-17 combined.
- United remain unbeaten at home in matches played on Friday since April 1950, winning 15 and drawing six since losing 2-0 to Birmingham City under Matt Busby.