Newcastle started the game brightly but fell behind in the 30th minute thanks to Marcus Rashford's delightful solo effort, but a goal of equally sublime proportions from Allan Saint-Maximin restored parity for Steve Bruce's men before half time.
In the second 45, a well-taken Daniel James strike preceded a Bruno Fernandes penalty as the hosts eventually ran out comfortable winners on their own patch, and they remain 10 points behind leaders Manchester City after their triumph at Arsenal a few hours before.
Newcastle, meanwhile, remain in 17th - three points above the relegation zone - although Bruce was able to take some positives from his side's performance.
Despite the 14 places and 21 points separating the two teams in the Premier League table, the visitors started off on the front foot and it took only four minutes for Joelinton to force the first save out of David de Gea in the Red Devils' goal.
The Spaniard returned in between the sticks after Dean Henderson was given the nod for the Europa League, and De Gea did not cover himself in glory as an attempted clearance found its way to Joelinton, who teed up Jonjo Shelvey only for the Englishman to blaze his effort over on the 10-minute mark.
Bruce's side were far more threatening in the opening exchanges and did not have the look of a team who had lost six of their last eight, and United's only real half-chance in the 20 minutes came courtesy of Dan James, but a United player could not get to his cross before Newcastle keeper Karl Darlow in the 18th minute.
Miguel Almiron was particularly bright and tenacious for the visitors, and the playmaker managed to test De Gea with a speculative effort after 27 minutes, but the United number one was equal to it. However, for all of their promise in the opening exchanges, it was Solskjaer's side who ultimately broke the deadlock.
Struggling to create any chances as a cohesive unit, Marcus Rashford took it upon himself to nutmeg Emil Krafth before cutting inside and firing a shot past Darlow at his near post on the half-hour mark. The Magpies goalkeeper would have no doubt felt disappointed not to keep out the Englishman's effort - his ninth goal of the Premier League season.
Despite the setback, Newcastle did not lose their drive and determination, with Almiron once again at the heart of Newcastle's attacks as Allan Saint-Maximin was denied by another acrobatic save by De Gea, but the Frenchman refused to be denied a second time.
A short corner routine in the 36th minute found its way to Joe Willock, and after Man United had failed to deal with the Arsenal loanee's cross, Saint-Maximin brilliantly sidefooted the ball into the roof of the net. The hosts' lead lasted all of six minutes before Newcastle deservedly equalised.
After being involved in a minor fracas only a few minutes before, Fred saw his name in flashing neon lights as he attempted a spectacular long-range effort in the 40th minute, but it was no trouble for Karl Darlow in the Newcastle goal as the sides could not be separated in the opening 45 minutes.
Newcastle started the second half with the same amount of attacking intent as the tricky Saint-Maximin had a good chance to double his tally at Old Trafford on the 50-minute mark, but the Frenchman's shot landed in the grateful arms of De Gea after it found its way through several red shirts.
However, as was the case with the first half, the hosts took the lead arguably against the run of play. Daniel James repaid Solskjaer's faith in him in the 57th minute, as Nemanja Matic turned into the box and found the Welshman in acres of space on the right-hand side, and the Welshman needed no second invitation to lash home at the near post after the ball took a nick off of Bruno Fernandes.
Solskjaer's men had their tails up after regaining the initiative, with the much-maligned Anthony Martial threatening a third goal for the hosts as some brilliant individual skill allowed him to get shot away in the 60th minute, but his effort stung the palms of Darlow and the resulting corner came to nothing.
By this point, Joelinton had been forced off through an injury which he sustained in the first half, and Bruce's men were under the cosh despite their promising start to the second period. Ryan Fraser did try his luck from long range in the 68th minute, but his weak effort was comfortably collected by De Gea.
With 20 minutes to go, any hopes of a famous Newcastle win appeared to have dissipated and United had golden chance to make it three in the 71st minute, but Harry Maguire could not get on the end of Fernandes's flick following Luke Shaw's corner. However, Solskjaer's men would give themselves a two-goal cushion only four minutes later.
Matic found Rashford in the area and he gave Joe Willock the slip, and even though the Arsenal loanee appeared to pull out of the tackle, he caught his compatriot and could have no complaints about the penalty which was awarded. In a tale as old as time, Fernandes coolly dispatched from 12 yards for his 15th top-flight goal of the season.
In the 83rd minute, Maguire did get on the end of not one but two headers from a pair of Shaw corners, but his first effort was tipped over by Darlow before his second went harmlessly wide of the post, although United were almost home and dry by this point of the match.
With the hosts almost guaranteed to take all three points home in the dying embers of the match, there was time for Solskjaer to hand Shola Shoretire his Premier League debut as he came on as a 88th-minute substitute for Rashford, and while fellow substitute Jacob Murphy had an attempt in the 90th minute, his effort was always rising.
A late foray forward from the Red Devils allowed Shoretire to get on the ball, but there was no urgency for Man United to go all guns blazing for a fourth goal as the hosts survived an early scare to secure another vital three points.
Solskjaer's side now turn their attention to their Europa League last-32 second leg against Real Sociedad on Thursday evening, while Newcastle have six days to recover before they take on Wolverhampton Wanderers next Saturday.
MANCHESTER UNITED: De Gea; Wan-Bissaka, Lindelof, Maguire, Shaw; Matic, Fred; James (Mata 88'), Fernandes, Rashford (Shoretire 88'); Martial (Greenwood 70')
NEWCASTLE UNITED: Darlow; Krafth, Lascelles, Clark, Lewis; Willock, Hayden, Shelvey; Almiron (Gayle 79'); Joelinton (Fraser 55'), Saint-Maximin (Murphy 77')