An unfortunate mistake from Dean Henderson allowed the hosts to take the lead in the fifth minute, but quickfire strikes from Marcus Rashford and Anthony Martial ensured the Reds were ahead before the break.
Then, early in the second period, Rashford capped a stunning counter-attack to provide what proved to the game's decisive goal.
The Blades claimed a fortuitous second with just minutes left, which made the final stages somewhat uncomfortable, but Solskjaer's men clung on to kick off the busy festive schedule with a positive result and climb to sixth place in the table, with a game in hand on the five teams above.