Luke Shaw's opener and a brace from Odion Ighalo earned a 3-0 victory over the Rams to set up a quarter-final tie at Norwich later this month.
It was a disappointing night for Derby player-coach Wayne Rooney, United's record goal scorer, who was twice denied by Sergio Romero in his reunion against them on Thursday.
But United extended their unbeaten run in all competitions to nine games ahead of Sunday's derby visit of Manchester City and remain in the hunt for the FA Cup, Europa League and a top-four finish in the Premier League.
Solskjaer said: "Today was the highest priority because we're not going to pick and choose games or competitions. We have to go for everything, that's what it is to be at Manchester United".