The hosts took a commanding first-half lead thanks to goals from Richarlison and Gylfi Sigurdsson, before Lucas Digne and Theo Walcott turned the game into a rout in the second half.
So poor was the performance, Solskjær went over to the traveling fans after the match and held his hands up in apology.
United have now conceded 47 goals this season, their worst defensive tally in Premier League history — and there are still four matches left to play.
Defeat leaves United two points behind Arsenal in fourth place — the final Champions League spot — though the Gunners play their game in hand against Crystal Palace later on Saturday.
The loss to Everton, which sees Solskjær’s side now on a run of six defeats in eight games, was the first time United have conceded four goals in a Premier League game since October 2016.
In the build up to this game, Solskjær had abruptly stated that some of his players needed a “reality check” following a poor run of form which culminated in last week’s humbling 3-0 defeat to Barcelona.
If he had been hoping for a reaction him team on Sunday then United’s performance will have left him exasperated, as Everton took a 2-0 lead with less than half an hour played.
The Norwegian coach, evidently bitterly disappointed with the performance, hauled off Phil Jones and Fred at half time for Ashley Young and Scott McTominay.
But things went from bad to worse for the visiting side as Digne made it 3-0 within 10 minutes of the restart and Walcott put the icing on the cake 10 minutes later.
Since that improbable comeback away at PSG in the Champions League — a ninth away win in a row — United have now lost five consecutive games away from home.
Up next is the small matter of the Manchester Derby, as Solskjær’s side welcome Pep Guardiola and Manchester City to Old Trafford on Wednesday in the hope of derailing their Premier League title charge.