The United coach saw his team exit the Champions League quarterfinals on Tuesday with a 4-0 aggregate loss to Barcelona, and Solskjaer's season took a further hit at Goodison Park with goals from Richarlison and Gylfi Sigurdsson before the break, and Lucas Digne and Theo Walcott after it.
As a reward for his better-than-expected spell in temporary charge of United, Solskjaer accepted a three-year contract on March 28 after 14 wins in 19 games.
But this loss damaged sixth-placed United's bid for a Champions League spot. Everton is seventh, 15 points behind United.
The loss was United's sixth in eight games in all competitions and turned up the heat on Solskjaer and his misfiring players, whose fortunes have slumped since the Norwegian was made permanent manager.
United has now conceded 48 goals in the league this season, their most in a top-flight campaign since 1979.
Romelu Lukaku, restored up front for United against his former club, struggled to get into the game and Paul Pogba did not impose himself in midfield.
Everton applied all the early pressure, winning a succession of corners and free kicks that allowed them to create problems in the United box.
Thirteen minutes had gone when Digne launched a long throw into the area and Dominic Calvert-Lewin flicked on. Richarlison met it with an acrobatic volley that flew into the net.
United got some relief when Pogba launched a long ball from inside his own half and Marcus Rashford raced in on goal but the England forward's half-volley flew narrowly over. Rashford also blasted a free kick over but Everton soon regained control.
Sigurdsson doubled the lead from a quick counterattack in the 28th. Nemanja Matic backed off enough to allow Sigurdsson to cut inside and unleash a fierce low drive from 30 yards (meters). David De Gea was slow to react and the ball shot into the bottom corner.
The interval failed to re-energize United, which conceded again 11 minutes after the restart.
Everton had suffered a blow by losing Richarlison to injury but it did not dampen the threat and Digne lashed in a long-range shot after De Gea punched clear from a corner.
Walcott, who replaced Richarlison, added the fourth in the 64th as Everton was again allowed to break all too quickly. He was released by Sigurdsson and showed his composure as he closed in on goal to slip the ball past De Gea.
United fans voiced their support for Solskjaer but the team's form is a serious concern with a tricky derby against Manchester City at Old Trafford on Wednesday and a home clash with Chelsea next Sunday.
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