United have lost six of their past eight games in all competitions and are sixth in the Premier League.
“From the first whistle, everything went wrong,” Solskjaer told Sky Sports.
“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.”
Speaking to Match of the Day, Solskjaer added: “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.”
Goals from Richarlison, Gylfi Sigurdsson, Lucas Digne and Theo Walcott handed United their biggest defeat of the season in a game where the visitors did not register their first shot on target until the 86th minute.
United are next in action against local rivals Manchester City on Wednesday.
“We just didn’t perform. That was not worthy of a Manchester United team,” said the Norwegian, who was appointed permanent manager in March.
“That performance is not good enough for a Manchester United team, from me to players, we let the fans down, we let the club down. That performance is difficult to describe because it is so bad.
“I want my team to be the hardest working team in the league, that is what we were under Sir Alex [Ferguson] – [Ryan] Giggs, Becks [David Beckham], Gary Neville, Denis [Irwin] – no matter how much talent you have got, you run more than anyone.
“You can’t change your whole squad. One step at a time. I am going to be successful here and there are players who won’t be part of that.”