Gary Rowett's men burst forward in search of an equaliser and were caught in stoppage time on the break when Anthony Martial combined with Lukaku to double United's advantage.
Derby finds itself in good position to secure promotion to the Premier League, as the Rams sit second in the Championship with 49 points. That's 11 goals this season for Lingard, and another beauty.
The Red Devils had more options up front, but their best chances were still falling to Rashford, who squandered a pair of opportunities after being put through by Pogba and Lingard, respectively.More news: Pakistan summons United States ambassador after Trump's angry tweet
Derby County made a very strong start to the second-half and were clearly growing in confidence; but still living dangerously as Mata, Lukaku and Pogba all had shots saved, blocked and fired narrowly wide respectively five minutes after the restart.
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.
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.
Marcus Rashford was the guiltiest party in the first half with a poor point-blank header that he tamely nodded against the post.