Anthony Martial’s opener and Bruno Fernandes’ second-half penalty were enough for United to hold on and beat Aston Villa, who had earlier equalised through Bertrand Traore.
And while Solskjaer refused to get carried away with his side’s position in the table, he believes their ‘immense’ improvement in the past 12 months means they can be counted as contenders.
‘I am very happy with the position we find ourselves in,’ admitted the United manager. ‘I think the league is going to be so tight. Every time you win a game, you take it one step further.
‘You can lose the chance to be in the title race in the first 10 games. So this is a good position to be in. If you can win games like tonight, it will make a big difference.
‘Of course we are happy with what we are doing. It is such a big difference to 12 months ago. We have improved immensely. We are getting fitter and fitter and stronger and stronger.
‘We should not get too carried away. It is not a cup final yet. But these points are important.’
United are now 10 games unbeaten in the league but Solskjaer believes they can still improve – particularly in seeing games out more comfortably.
‘We need to get better,’ added Solskjaer. ‘I did not enjoy last 10 minutes. It seemed like it was a cup final or final game of season. We need to learn how to see game out better.
‘There’s a reason for the colour of my hair (grey) – and that’s because we like to do it the hard way.’
Aston Villa boss Dean Smith, who said his side played like it was a ‘testimonial’ in the first 10 minutes, was left to rue the awarding of Fernandes’ match-winning penalty.
Referee Michael Oliver did not consult his pitchside screen after Paul Pogba went down in the box under a challenge from Douglas Luiz, even though the Frenchman appeared to trip himself up.
Smith said: ‘We lost to a dubious penalty. It looks like he has tripped himself up. There is very, very minimal contact.
‘For me, Michael should have been sent to the camera. That is what VAR is for. If there is so much doubt then get yourself over there. I am a little bit dumfounded as to why he wasn’t sent over to have a look at it.’