Lingard and Paul Pogba both had early attempts for a United team that started on the front foot at Old Trafford. Pogba was the one making things happen down the left, and the Frenchman just missed the crossbar with a rasping strike in the eighth minute as the home side continued to press early on.
Marcus Rashford was next to come close for United in the 18th minute after driving into the Derby box, but the attacker just missed the crossbar, before Derby goalkeeper Scott Carson kept out a curling free kick from Juan Mata.
Marcus Olsson had Derby's first chance in the 22nd minute, and Sergio Romero had to be alert to tip over the crossbar. Sam Winnall then dragged one wide of the United post just before the half-hour mark as the visitors started to enjoy themselves.
Rashford headed against the Derby post in the 34th minute, before heading straight at Carson three minutes later as the score somehow stayed goalless.
Carson then made a flying save to keep out a Pogba free kick in the final stages of the first period, which finished goalless.
Lukaku came off the United bench at the interval as Mourinho looked for more cutting edge in the final third of the field. It was Derby that had the first half-chance of the second half, however, with Tom Huddlestone seeing his strike deflect just wide.
Pogba then curled one wide of Carson's post moments later as both teams had opportunities after the restart. Derby were not afraid to continue pushing forward, and Romero had to be alert to keep out a strike from Lawrence just before the hour mark.
United were the team pressing as the match headed for its final 20 minutes though, and Rashford struck one against the woodwork in the 71st minute after working space in a wide angle.
Pogba then had a golden chance in the 77th minute after the ball fell for the Frenchman inside the Derby box, but he scuffed his effort wide, and it did appear that the tie was heading for a replay.
Lingard effectively won it for United in the 84th minute, however, when the Englishman netted his eighth goal in his last 10 appearances - finding the top corner from distance to break Derby's hearts in the latter stages.
Lukaku, who showed no ill-effects following his return from a head injury, then doubled the home side's lead in the final moments when he converted after smart work from fellow substitute Anthony Martial.
UNITED (4-3-3): Romero; Lindelof, Smalling, Blind, Shaw; Herrera, Pogba, Mata (Martial 68'); Lingard, Rashford (Fellaini 80'), Mkhitaryan (Lukaku 45')
DERBY (4-5-1): Carson; Wisdom, Keogh, Pearce, Olsson; Huddlestone, Thorne, Russell (Hanson 78'), Lawrence (Bennett 81'), Weimann; Winnall (Vydra 68')