The in-form Championship side looked set to hold out but Lingard's eighth goal in 10 games in the 83rd minute put United ahead and Romelu Lukaku's stoppage-time goal sealed it
It had looked like being a frustrating night for Jose Mourinho's team as Marcus Rashford hit the post twice and Scott Carson pulled off a series of fine saves for Derby
But United deserved their win despite the visitors' brave efforts to keep them out and go into the fourth round with hopes of lifting the trophy they last won in 2016
The home side began brightly and almost all of the attacking players tried their luck in the opening stages, Ander Herrera heading wide and Juan Mata forcing a good save from Carson
The breakthrough really should have come 10 minutes before the interval when Rashford headed Henrikh Mkhitaryan's right-wing cross against the post from close range
Mkhitaryan's form remains a concern, surrendering possession sloppily at times, and he was replaced by Lukaku at the interval
 By that point, Paul Pogba had tested Carson with another free-kick that the goalkeeper did well to tip wide
Derby were only an intermittent threat on the counter-attack through the powerful running of former United man Tom Lawrence but they got to the break with hopes of a replay intact
That looked likely when Rashford's right-footed shot on the angle struck the foot of the post and bounced clear with 20 minutes remaining before Pogba missed another good chance soon after
But Lingard, in the form of his life, broke the deadlock against the club he was on loan at in 2015 by finding the far corner from just outside the box to the delight of the Old Trafford crowd
There was still time for Lukaku to combine brilliantly on the break with fellow substitute Anthony Martial and mark his return to the squad with a goal, beating Carson with his right foot
It was cruel on Derby but United deserved their place in the fourth round
"I think we had a good game
 They gave us a difficult game because they defended very, very well
 But we deserved the victory
 To be honest, I don't think we need a replay
 I don't think a team like them fighting for promotion needs a replay but that's the rule of the competition and we almost had to go for a second game
" "We started a little bit tentatively, but the key for us really was to go and enjoy it, there was no pressure on us
It was important to see which players could handle it because of course we are aspiring to be playing games like this every week next season
"Lingard has scored only 11 goals in his Manchester United career prior to this season but he has now doubled that total thanks to his goals in the current campaign
 It is eight goals in his last 10 games and this was yet another decisive late goal - and a wonderful hit at that
Manchester United are due to host Stoke next time out in the Premier League on Monday Night Football on January 15th
 Derby's focus will now return to the Championship and they travel to Birmingham on Saturday January 13th
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