Solskjaer's side were never in danger of a shock fifth round exit against the second tier Rams as Luke Shaw's opener and Odion Ighalo's double booked a quarter-final trip to Norwich.
Rooney, United's record goalscorer, is back in English football after moving to Derby from DC United in January and he was twice denied by Sergio Romero in his first meeting with his old club since 2018.
But United were always on top and extended their unbeaten run in all competitions to nine games ahead of Sunday's derby showdown with Manchester City at Old Trafford.
They remain in the hunt for the FA Cup, the Europa League and still have a chance of qualifying for the Champions League via the Premier League.
United are fifth, three points behind Chelsea, and Solskjaer has urged his players to keep their foot on the gas as they look to turn a troubled season into a memorable finale.
"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," Solskjaer said.
"It's another away draw, we're use to that now. Is it a seventh in a row? We'll have to do it the hard way if want to go all the way. We've got a chance now.
"I'm very pleased with the performance and to go through. You never know, away from home, being the favourites and having City on Sunday how it'll affect their mindset, but it was professional.
"We've found form and defended well, we've had seven clean sheets in those nine games."
United are chasing a record-equalling 13th FA Cup triumph.
They took control at Pride Park after 33 minutes when Shaw scored just his second career goal, his volley bouncing over Kelle Roos via a deflection off Jesse Lingard.
Nigerian striker Ighalo, signed on loan from Shanghai Shenhua in January, got the second goal seven minutes later.
Martyn Waghorn went close for Derby after the break but Ighalo ended the contest when he fired home to complete his double with 20 minutes left.
"He's doing what he says on the tin and what we asked for when we signed him. I'm very happy with him. That type of striker is vital for a team.," Solskjaer said of former Watford forward Ighalo.
"There's more to come. He'll improve and get sharper and more used to us. I've got no doubt he'll only get better."
Former England star Rooney, who left United to rejoin Everton in 2017, almost grabbed a consolation but Romero produced an excellent stop to turn his late free-kick over.
Rams boss Phillip Cocu said: "If the first free-kick goes in, sometimes it defines a game.
"You have to be really effective and on that side they were much better than we were.
"I'm quite satisfied with the performance."
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