Almost 25 years to the day since Matt Busby died, Solskjaer made it seven wins from seven in all competitions on the one-month anniversary of his appointment as two of the players to have shone in his short tenure scored again.
Paul Pogba converted a 27th-minute penalty he won himself for his fifth goal in his last three games at Old Trafford before Marcus Rashford marked his 150th appearance with the second prior to the interval.
Pascal Gross pulled one back for Brighton, but United claimed the three points/\.
Rashford and Pogba had already threatened the Brighton goal before United was awarded a penalty.
Pogba latched onto a long Nemanja Matic pass on the right flank by taking a touch inside Gaetan Bong, who bundled into the back of the Frenchman to concede the spot-kick.
The usual stuttered run-up followed but Pogba placed his effort into the right-hand corner, past David Button's right hand to put United in front.
Button was able to make himself big and prevent Anthony Martial making it 2-0, while Brighton's Glenn Murray fired a great chance wide with a miss that would look even more costly before the break.
With three first-half minutes remaining left-back Diogo Dalot, a late inclusion in the starting XI due to Luke Shaw's injury, fed Rashford and he cut inside Gross before bending a fine right-footed finish into the far corner.
Rashford continued to look in the mood after the interval, firing in a cross that Jesse Lingard somehow failed to direct home from close range.
Martial blasted into the side netting but United's intensity dropped and Brighton were given hope with 18 minutes to go.
Davy Propper's outside-of-the-foot cross reached Gross, who chested the ball down and poked the ball in off the underside of the crossbar.
A nervy finish for the home side therefore ensued and though Pogba was unable to alleviate the tension when rifling just wide, United held on to round off a fantastic first month in the job for Solskjaer.