Pogba was off target with a ninth-minute curler while Rashford was similarly wide from an ambitious effort with 17 minutes on the clock.
The 25-year-old was named United's player of the month for December, and scored the second goal for them in the 2-0 victory at Everton on New Year's Day.
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.
Tom Lawrence almost sent Derby ahead two minutes before the hour with a thumping shot from just outside the box but Sergio Romero made a brilliant save at the bottom corner.
Rashford nodded against a post after being set up by Mkhitaryan ten minutes before the break, before heading straight at Carson - this time from a Mata delivery. It was Derby that had the first half-chance of the second half, however, with Tom Huddlestone seeing his strike deflect just wide.
"But when you get past 80 minutes and the goal hasn't come, you fear the second match".
The breakthrough would eventually come on 84 minutes, with Lingard unleashing a cracking effort from outside the box after a lay-off from sub Romelu Lukaku.
Lukaku was next to head wide on 55 minutes, before Thomas Lawrence forced a diving save from Romero.
Marcus Rashford was the guiltiest party in the first half with a poor point-blank header that he tamely nodded against the post.