United twice led but missed the chance to go second after Jamie Vardy's deflected strike off Axel Tuanzebe kept the Foxes above their opponents.
Marcus Rashford's strike in the 23rd minute was cancelled out on the half hour mark by Harvey Barnes before Bruno Fernadnes looked to have scored the winner in the 79th minute.
But the United defence switched off with five minutes of normal time remaining with Vardy hitting the ball off Tunazebe's legs and into the back of the net.
'We feel disappointed. The boys are flat in the changing room,' said Solskjaer after the game.
'Great movement by Vardy. You can't switch off against quality players. We created big chances. It was a good game of football, end to end at times.
'Both goals should have been closed down quicker. The first one definitely and second one down the right-hand side we should stop the cross. You always look at the goals and we could and should have done better.
'We have a game in three days and we got an extra recovery day which we will use wisely. One point is not the worst result but we're disappointed as it's a good side we're up against.'
Fernandes, who grabbed his 10th league goal and sixth assist of the season, insisted he and his team-mates were not satisfied with a point and praised the 'winning mentality' they are creating at the club.
'I was really happy when I got in the dressing room and I saw everyone disappointed and saying the things we had to do better,' said the midfielder.
'You are creating a winning mentality and we are not happy with this point.'
The draw keeps Leicester in second with 28 points, one ahead of United, who stay third with a game in-hand.
The Foxes travel to Crystal Palace and Newcastle for their next two fixtures while United host Wolves and then Aston Villa.
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