United was beaten by a double from in-form Kelechi Iheanacho and one from Youri Tielemans at the King Power Stadium as the Foxes secured just a second victory over the Red Devils in 26 meetings in all competitions.
Mason Greenwood's first-half leveller – his first goal in 16 games – ended up being a mere consolation for the visitors as Brendan Rodgers marked his 100th game in charge by leading Leicester to their first FA Cup semi-final since 1982.
For United, it was a first away defeat in domestic competition since 20 January last year, with manager Solskjaer citing a busy fixture list that has recently included a tough two-legged Europa League tie with Milan.
"We didn't have the spark, but it's understandable," he said. "This team have been fantastic in the last three or four months. We played every three days and have been on a great run. It just caught up with us, all the games and travel.
"Thursday night in Milan was a big night and took a lot of out of us physically. We didn't have the extra zip, authority and confidence.
"We tried to get on the front foot and start well because sometimes you can get that adrenaline boost and confidence if you get a goal and that can carry you over the line.
"We've got too many players that have played too many games and too many that haven't really had much football; Anthony Martial, Donny van de Beek, Paul Pogba and others are trying to get in who haven't had enough football. We didn't have enough against a tough team."
The first goal came about after a woeful mistake from Fred, whose attempted pass back to Dean Henderson gifted former Manchester City striker Iheanacho a simple finish.
But the Norwegian refused to point the finger at any individuals, insisting such mistakes come with the territory of playing out from deep.
"We want to build from the back and want the players to be confident," he said. "Sometimes it doesn't work and that's one time it didn't.
"When we win, we win together, and when we lose, we lose together. We're not pointing fingers and blaming.
"The second one is a good goal by the boy but maybe with a zip of energy we could've got to him and wouldn't have been done as easily in the middle.
"That's been one of our strengths, keeping clean sheets and being dogged, but we didn't have the opportunity."
Solskjaer is now looking ahead to what remains of United's season, which still holds the prospect of continental silverware and a shot at sealing second spot in the Premier League.
"We've got the Europa League and Premier League to concentrate on," he said. "Yes, we would've liked to go to Wembley, but now all the focus is on the games we do have.
"We're in a good position in the league and we want to keep improving; of course, Leicester are just behind us, so it won't be easy, but we want to keep going and get as far as possible in the Europa as well."