Solskjaer's side added to Liverpool's woes as Fernandes capped a thrilling tie with his late strike at Old Trafford.
Premier League leaders United recovered from Mohamed Salah's early opener to score through Mason Greenwood and Marcus Rashford.
Salah equalised before Portugal playmaker Fernandes came off the bench and underlined his class with a superb winner in the 78th minute.
"Great goal. When you leave him out like I did today, he stayed about 45 minutes after training yesterday shooting free-kicks so I was pretty confident he could score one if he got the chance," Solskjaer said.
"He's never happy with me when I tell him to go inside after training when he's playing the next day so he got some practice yesterday."
United, who sit six points clear of Liverpool in the league, are on a superb run after shaking off their poor start to the season.
Coming from behind to beat a team of Liverpool's quality was further proof of their renaissance.
"Brilliant. You know when you go a goal down, the reaction of everyone was really good," Solskjaer said.
"We played some really good stuff, good goals, we have to defend well against them and we managed to react well.
"Of course towards the end there were some moments but we kept them away.
"We felt if we could switch the ball early we could attack them because Liverpool defend on the front foot, so we had to be quick on the ball. We did that well."
Since his United debut in February 2020, Fernandes's 28 goals are more than any other Premier League player.
"I'm happy because we won, nothing more. Most important was going to the next round. It doesn't matter who scores. Of course if I score, I'm happy, that's for sure," Fernandes said.
"I work a lot on these kind of goals, free-kicks. I tried, this one was good.
"Before the free-kick, Edinson Cavani told me, 'Try the hard one on the side of the keeper'. I tried it and it went well."
Asked how excited he is by United's recent progression, Fernandes said: "A lot. As I said when I had just arrived this is everything, a dream come true.
"To play for this big club, to play in the Premier League, it's a dream come true."