Derby finds itself in good position to secure promotion to the Premier League, as the Rams sit second in the Championship with 49 points.
Anthony Martial combined well with Lukaku on the counter in stoppage time to ensure of the victory and make it two games, two wins and two clean sheets to kick-off 2018.
The visitors had a rare chance three minutes before the break as Marcus Olsson fired a shot from just outside the box following a corner but it sailed over the bar and the half ended goalless.
The half's final opportunity asked for another top Scott Carson save.
Manchester United had the first attempt on goal through captain Pogba firing a straight shot at Carson in the second minute. However, Zlatan Ibrahimovic, Eric Bailly and Antonio Valencia will be unavailable because of injury issues.
United also got good news regarding Romelu Lukaku, as the Belgian global will be fit and available to play after suffering a scary blow to the head against Southampton on December 30.
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Rashford hit the left side of post with a right-footed shot in the 70th minute from the right side of the box after neatly inter-changing passes with Lingard.
United await Monday's draw to discover who they will face in the fourth round.
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.
It looked set that Mourinho's men were going have have to travel to Pride Park for an unwanted replay with seven minutes to go, but the hottest man now on the block at Old Trafford had other ideas.
Rashford's finishing let the hosts' approach play down, though, when the striker skied over the bar after being played through.
Derby Team to play Manchester United: Carson; Wisdom, Keogh, Pearce, Olsson; Thorne, Huddlestone; Weimann, Russell, Lawrence; Winnall.