Well-taken goals from Richarlison and Gylfi Sigurdsson had the Toffees 2-0 up at the break before Lucas Digne's sweetly struck effort and Theo Walcott's cool finish completed a memorable win.
United are two points off fourth but will be three adrift if Arsenal overcome Crystal Palace and the inquest will surely soon begin after a nightmare outing that strikes a huge blow to their hopes of featuring at Europe's top table next season.
8 - Gylfi Sigurdsson has been directly involved in eight Premier League goals against Manchester United (five goals and three assists), more than he has versus any other club in the competition. Dagger. #EVEMUN pic.twitter.com/WL0hi3wm0B— OptaJoe (@OptaJoe) April 21, 2019
Richarlison had already brought a fine reaction stop from David De Gea on the bounce by the time he opened the scoring.
Dominic Calvert-Lewin flicked on Lucas Digne's header and the Brazil forward lashed home an unstoppable acrobatic volley.
Marcus Rashford flicked over the top from Paul Pogba's long punt but United's situation worsened after 28 minutes.
Everton broke from a United corner, leading to Sigurdsson cutting inside from the left and drilling low into the near corner from 25 yards.
The game was all but done 10 minutes after the break when Digne showed sensational technique on the half-volley to leave De Gea no chance.
Further humiliation was to follow and Sigurdsson carved United open again to play through substitute Walcott, who slotted past De Gea one-on-one to cap off a day to forget for Solskjaer and his team.
Wheels fallen off for Ole?
'Ole's at the wheel' was the chant from the United fans during Solskjaer's sensational start to life in the dugout at United. But six defeats in eight competitive outings and three straight away league losses for the first time since January 1996 mean the honeymoon period is well and truly over. Out of the Champions League and possibly out of the race for fourth, Solskjaer needs a solution and fast.
Sigurdsson shines in the sun
As bad as United were, Everton deserve credit for running roughshod over them on a glorious Easter Sunday on Merseyside. Sigurdsson was at the heart of the Toffees' play and finished with a goal and an assist in a performance that was arguably the best they have produced under Marco Silva.
United's performance bordered on embarrassing and singling out the worst showing is no easy feat. But Phil Jones had a particularly difficult afternoon, being dragged from pillar to post during a woeful defensive showing from the Red Devils, who have now conceded their most ever goals in a Premier League campaign.
Key Opta Facts
- This was Everton's biggest margin of victory over United in any competition since October 1984, when they beat them 5-0 in a top-flight contest. It was also United's biggest defeat to any opposition since October 2016 against Chelsea in the Premier League (also 0-4).
- This was United's fifth consecutive away defeat across all competitions, their longest such run of losses since March 1981 under Dave Sexton (also five).
- United have conceded 48 Premier League goals in 2018-19, their highest ever tally in a single season in the competition and their most in a league campaign since 1978-79 (63).
- Everton have won more Premier League points at home to the "big-six" clubs this season (10 – P6 W3 D1 L2) than any other team from outside the top six in the table, with only Man City (15), Liverpool (13) and Chelsea (11) earning more overall at home to such opposition.
- United have gone 11 consecutive matches without a clean sheet in all competitions for the first time since December 1998.
- Only Andrei Kanchelskis (15 in 1995-96), Yakubu (15 in 2007-08) and Romelu Lukaku (15 in 2013-14) scored more Premier League goals in their debut season with Everton than Richarlison has this term (13).
Everton face a tough trip to Crystal Palace next Saturday, while there is a small matter of the Manchester derby for a despondent United when City visit Old Trafford on Wednesday.