Goals from Richarlison, Gylfi Sigurdsson, Lucas Digne and Theo Walcott inflicted United's sixth defeat in eight games in all competitions, with the lacklustre manner of this latest reverse a cause for concern.
"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 heads high, because we can't."
The former United striker added: "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)."
But he insisted sixth-placed United could transform their fortunes before the end of the season.
"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 can turn it around, in football like in life things can change quickly. We have got to change it from bad to good."
Gary Neville, a former United team-mate of Solskjaer has long been critical of the club's hierarchy for the way in which a succession of managers have come and gone since Alex Ferguson retired in 2013.
He did not spare the "rancid" team on Sunday.
"The United players, it's embarrassing watching some of them out there today," Neville said.
"I'm absolutely furious. Full belief in Ole to do what he needs to do, the club will obviously support him.
"But there's a big problem and I have to say that performance was rancid."
Everton, who had not beaten a 'big six' rival in 25 attempts before last month, have now won consecutive home matches against Chelsea, Arsenal and United while keeping three clean sheets in the process.
"Delighted with the players, it's important to remind you what we did in the last four home games -- we're on a fantastic run," said Everton manager Marco Silva after a win made all the more impressive after the way his side lost 2-0 at already-relegated Fulham last weekend.
"Our last performance away was not good enough and we must show something different, we did," he added. "We were the best team on the pitch from the first to last minute."
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