Three goals in the first half from Marcus Rashford, Ander Herrera and Anthony Martial and a brace from Jesse Lingard earned all three points at Cardiff City Stadium, with Victor Camaras's penalty proving little consolation.
United stays sixth in the table, but the manner of the performance, as well as the result — the first time the side has scored five times in the league since Alex Ferguson's final game — will have been a welcome sight for the travelling fans at the end of a week that saw Jose Mourinho's contentious time in charge come to an end.
The new United era got off to a stunning start when Rashford blasted home a free-kick from 30 yards out that left goalkeeper Neil Etheridge rooted to the spot.
Almost 81 per cent of possession underlined United's early control, and the scoreline reflected their dominance when Herrera's strike looped off Greg Cunningham's shoulder and into the net 29 minutes in.
Cardiff was gifted a lifeline when Camarasa fired high into the net from the spot, but United's two-goal lead was restored three minutes later, when Martial slotted beneath Etheridge after a sublime one-touch passing move involving Lingard and Paul Pogba, restored to the starting line-up.
Any hope of a Cardiff comeback was promptly extinguished 11 minutes into the second half, when Lingard dispatched a confident penalty after he had been bundled to the ground by Sol Bamba.
Pogba and Phil Jones were each denied by good Etheridge saves, while David de Gea kept out a good Kenneth Zohore attempt.
It was 5-1 when Lingard rounded the keeper and slotted home from Pogba's pass in the 90th minute to secure a first away league win since 4 November )AEDT) in style and give Solskjaer extra cause for celebration against the club he used the manage.