Ole Gunnar Solskjaer walked straight over to the Manchester United fans after their and held up his hands in apology.
United were way below par at Goodison Park, conceding two goals in each half on a damaging day for their hopes of finishing in the top four.
It took the Reds until the 86th minute to have their first shot on target, with former United captain Gary Neville describing the performance as "rancid".
Solskjaer was joined by Scott McTominay and Diogo Dalot in staying on the pitch long after the final whistle to recognise the travelling Reds at Goodison Park.
The Norwegian's name was sung defiantly by the United fans in the final half hour of the game and he later thanked the fans for how supportive they were in trying circumstances.
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Solskjaer told the BBC: "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 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."
United will turn their attention to a Manchester derby clash against Man City on Wednesday night, with Solskjaer and the players needing a reaction.
The boss said his players "need a game of football as soon as we can to get this out of the systems".
He added: "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."
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