Shaw opened the scoring for United after 33 minutes with a shot that deflected off Lingard and looped over Roos.
Odion Ighalo then took over with a couple of goals and should have scored a hat-trick, only to fire over from a good position late on.
Solskjaer’s men are now unbeaten in nine matches, winning six of those games, ahead of Sunday’s Premier League showdown with bitter rivals Manchester City at Old Trafford.
As for Derby, they have three days to pick themselves up before hosting Blackburn Rovers on their return to Championship action.
🎶 "Olé, Olé, Olé, Olé" 🎶
— The Emirates FA Cup (@EmiratesFACup) March 5, 2020