Sanchez, Jesse Lingard and Anthony Martial all scored in a counter-attacking masterclass to send United marching into the FA Cup fifth round.
It’s now eight wins in a row for their interim boss, and even a full-scale melee towards the end when at least one missile was thrown from the crowd could not take the gloss off it for Solskjaer.
He said: “It was a massive step forward. 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. It’s easy to get excited and talk about the future but we need to just keep working and think about the next game”
Ole Gunnar Solskjaer
“Today was a big step forward. It was a massive improvement. 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. It’s fantastic to go through in the cup. It’s one of the hardest games you can get.
“I’ve had a headache 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.”
Solskjaer also hailed the contribution of Sanchez, who scored the opener, saying: “It was a great goal. He has been injured and it was great to see. He can only improve.”
Arsenal, who had won the cup in three of the past five seasons, crashed out despite Pierre-Emerick Aubameyang pulling one back.
They also lost Sokratis and Laurent Koscielny, who was rushed to hospital with a suspected broken jaw, to injuries too.
Gunners boss Unai Emery said: “Our injuries changed the possibility to do something different on the pitch. Sokratis hurt his ankle – I hope not a lot. Koscielny is different. It was maybe a break.
“I hope not but he’s going to the hospital and we’re going to wait. We are taking a lot of big injuries, like Hector Bellerin.
“We played a team in a good moment. We lost this match but I’m happy with our work. The supporters helped us.
“We need to think about the Premier League. We won against Chelsea and are close. We need to push in that competition and we have another important competition in the Europa League.”