United was brilliant in the first half, as impressive as it has been in the 12 months of its Norwegian manager's tenure, and a penalty from Marcus Rashford followed by Anthony Martial's sharp finish made it the Red Devils' day. A late Nicolas Otamendi header could not save the host.
City sits 14 points behind Liverpool in the Premier League, hopes of defending its title surely all but extinguished by this result. Questions will be asked of Pep Guardiola's leadership, and how he has allowed the defence to become such a weakness.
But for Solskjaer, months of speculation about his job security can be swept aside for now. After beating Jose Mourinho's Tottenham and following up with a terrific performance against its 'noisy neighbours', United suddenly looks a team to be reckoned with.
United's first three attempts all forced Ederson to make sharp saves, the goalkeeper denying Daniel James from the edge of the penalty area, then a lively Jesse Lingard and the marauding Martial.
And with City in disarray at the back, the fast-breaking visitors were handed a penalty after a VAR review when Rashford went down following a mazy run, sideswiped by a clumsy Bernardo Silva challenge.
Rashford slotted low and into the right corner for the 23rd-minute opener, and the England striker went close twice in the next four minutes: firstly, when stroking a shot just wide after City's defence was unpicked by a high-pace counter-attack, and then with a 20-yard curling shot that hit the crossbar.
Martial gave United the two-goal lead they richly deserved in the 29th minute when he swivelled and cracked a low, left-footed shot in off the base of the right post, after working the chance patiently with James.
City striker Gabriel Jesus might have narrowed the lead before half-time but headed wastefully wide from Kevin De Bruyne's cross, before the hosts were aggrieved to be denied a spot-kick by referee Anthony Taylor when Fred appeared to handle a cross.
Guardiola made his annoyance known in a passionate appeal to fourth official Mike Dean before the second half began.
United centre-back Victor Lindelof made a superb intervention to thwart De Bruyne soon after the restart, deflecting the Belgian's thumping shot over the crossbar.
Fred was struck by an object thrown from home supporters and appeared to suffer racist abuse from the same section.
Otamendi headed in an 85th-minute City consolation and fellow substitute Riyad Mahrez went close to a leveller moments later, denied by a fine save by David de Gea, but this was emphatically United's day.