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 Red Devils have already lost to a Championship club this season, as they were dumped out of the Carabao Cup by Bristol City in December, and they could not be facing a more in-form side, as Derby have only lost one of their last 11 matches. But Sam Winnall failed to profit as he dragged his shot wide.
The much-changed Rams held their own for most of the opening 20 minutes, despite some fine play from Paul Pogba and Juan Mata in the United midfield.
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.
Lingard, in the form of his life, pulled off a stunning first-time effort from outside the area to break the deadlock on 84 minutes.
Romero, Lindelof, Smalling, Blind, Shaw, Herrera, Pogba, Mata, Lingard, Mkhitaryan, Rashford.
"But when you get past 80 minutes and the goal hasn't come, you fear the second match".
Rashford's finishing let the hosts' approach play down, though, when the striker skied over the bar after being played through.
Lukaku was next to head wide on 55 minutes, before Thomas Lawrence forced a diving save from Romero.
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.