Solskjaer stated: "From the first whistle everything went wrong. I just want to apologise to the fans. They're the only people with the badge on today who can hold their head high, because we can't.
"Everton were better than us at all the basics. There are so many things that we need to do better to get a result. Talent has never been enough.
"We just didn't perform. That was not worthy of a Manchester United team. We hold our hands up and apologise to everyone associated with the club. We have a perfect chance to make amends on Wednesday [against Manchester City]."
The Norwegian also stated: "We were beaten on all aspects today and the only place we beat Everton - I can't say we beat them - but we had a fantastic support and I just want to apologise for the performance we turned out.
"We have been fantastic for long periods and it's a tough league to get into the top four, never mind winning the league - there are six teams there and Everton want to be in there.
"We know it's a big task and work to be done. We have to do it together, we have to get on with it because this is not good enough.
"You have got to start somewhere and we will give it everything we have got with the last four games - massive games with City and Chelsea coming to Old Trafford - that's where we get the best support in the world.
"We can turn it around, in football like in life things can change quickly. We have got to change it from bad to good.
"We need a game of football as soon as we can as we need to get this out of the systems, everyone here can say with hands on hearts that's not good enough, not worthy of a Manchester United team. We know that and apologise again to fans."