United made it eight wins on the spin after beating Arsenal 3-1 at the Emirates Stadium this evening.
Alexis Sanchez netted against his former side before Jesse Lingard doubled the lead soon after.
Anthony Martial then scored after the interval to bolster his chances of a start in the coming weeks.
And it now appears Solskjaer has a selection headache to deal with due to his attacking stars all knowing how to find the back of the net at present.
“It’s fantastic to go through in the cup. It’s one of the hardest games you can get,” he told BBC Sport.
“I’ve had a [selection] headache since I came because the squad is full of good players.
“It’s up to us to pick the right tactics and players. We need to rest a few as there’s big games coming up. But we have fresh legs.
“Today we looked more like a proper team. We looked more structured.
“We’d worked hard at defending and counter-attacking. Counter-attacking is classic Manchester United – Wayne Rooney, Cristiano Ronaldo, Ji Sung-park – we’ve scored so many at Arsenal over the years.
“Today was a big step forward. It was a massive improvement from the Tottenham game.
“It’s easy to get excited and talk about the future but we need to just keep working and think about the next game.
“We had to dig deep. The attitude and work-rate was important. Towards the end it was comfortable but not during the 90 minutes.”
Speaking after clash, Romelu Lukaku added: “[It looks like fun to play in this team?] It does it does, We had clear direction today and I’m really happy with the win and the manner in which we played.
“I just take it a day at a time. I work hard in training and it paid off today and we own.
“Now it’s up to me to keep working and help the team with goals and assists.
“The guys just have to put the ball in the back of the net!
“I always wanted to be a player who can do both, score goals and get assists.
“Look at the history of the club, we make a statement wherever we go. We look to dominate and create as many chances as we can.”