Defeat at Goodison Park -- United's sixth in eight games -- also dealt a huge blow to the Red Devils' hopes of playing in the Champions League next season.
Goals from Richarlison, Gylfi Sigurdsson, Lucas Digne and Theo Walcott inflicted the largest defeat on United since they lost by the same scoreline at Chelsea three years ago.
But, after starting his reign with eight consecutive victories, the best start ever made by a manager at the club, former United striker Solskjaer is now collecting some other, very unwanted, statistics.
This latest loss means United have now leaked 48 league goals, the most they've conceded in a season, since letting in 63 back in the 42-game 1978/79 campaign.
Sunday's loss also means United have conceded in 12 consecutive games for the first time since December 1998 while, under Solskjaer, the Reds have lost five consecutive away games for the first time in 38 years.
United looked second best even before the hosts scored twice inside the opening half-hour, conceding a succession of corners from which Everton threatened.
It was another type of set-piece, a long throw-in by Digne, from which Everton took the lead after 12 minutes when the full-back launched the ball deep into the United area.
Dominic Calvert-Lewin helped the ball on with the flick of his head and Richarlison sent a spectacular scissor-kick finish flying past David de Gea.
After a humbling 3-0 Champions League exit at Barcelona in midweek, it was not the start Solskjaer had been looking for. Everton soon made the situation worse for the visitors.
After 27 minutes, Everton cleared a United corner and immediately broke on the counter-attack, with winger Bernard playing a delightful ball inside to Idrissa Gueye.
He, in turn, released Sigurdsson who, with Nemanja Matic backing off, unleashed a magnificent 25-yard shot which bounced just in front of the diving de Gea on its way into the bottom corner.
Richarlison might have given Everton the lead even earlier. His 10th minute shot drew an amazing reaction save from de Gea. From the rebound the Brazilian hurried his follow-up effort wide.
With the recalled Romelu Lukaku booed every time he touched the ball by his former fans at Everton, United had little response to an impressive effort from the home side.
In between the goals, United star Paul Pogba's long diagonal pass freed Marcus Rashford who could only poke his effort over from a tight angle.
Everton effectively killed the game in the 56th minute. De Gea punched clear a Sigurdsson corner to the unmarked Digne who sent a superb, first-time volley flying back past the United keeper from 20 yards.
Three minutes later, and from the same left wing, another Sigurdsson corner almost crept in at de Gea's near post before the Spaniard stuck out a boot to clear off his line.
Even though Richarlison had limped off through injury, the agony was not over for United with the Brazilian's replacement Walcott continuing the rout in the 64th minute.
Sigurdsson again carved out the opening with a brilliantly-timed through ball which allowed the former England international the space to sprint into the area and beat the advancing de Gea.
Not until the 86th minute, and a comfortably saved Anthony Martial strike, did United have a shot on target.
Meanwhile 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.
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