The Red Devils failed to extend their winning run away from home to eleven matches after the Foxes came from behind twice to snatch a crucial point.
Marcus Rashford's opener inside 23 minutes was canceled out by Harvey Barnes eight minutes later.
Bruno Fernandes' 79th-minute goal looked to have maintained United's perfect away record this season but Axel Tuanzebe deflected in Jamie Vardy's strike to level it up on 85 minutes.
"Disappointed we didn't win today because we had loads of big chances and we could've finished it off but they're a dangerous side, the effort from Jamie Vardy just shows his quality,” Solskajer said after the game.
"It was a good goal from Marcus Rashford as well and I'm glad he scored. He knows the first chance was a glorious one [missed chance in the first half]. Missing that is just one of those things.
"Both goals should've been closed down quicker - the first one definitely and the second one we should manage to stop the cross. You always look at goals conceded, what we should've or could've done better.
"Of course we've got a game in three days - we have an extra recovery day which we will use wisely. One point is not the worst result but we're disappointed to not get the three points against a tough side."
