Marcus Rashford smashed United into an early lead with the help of some poor goalkeeping by Paulo Gazzaniga as the England international’s powerful shot crept in at his near post and had chances to extend the home side’s lead.
Yet for all United’s dominance in the first 45 minutes, one moment of brilliance for Dele Alli got Mourinho’s men level before the break.
Alli plucked the ball out of the air and skipped beyond two United defenders with one touch before slotting home his fourth goal in as many games since Mourinho took charge.
United were handed the chance to retake the lead when Rashford went down under Moussa Sissoko’s challenge in the penalty area early in the second half and he put his recent spot-kick woes behind him to send Gazzaniga the wrong way.
“The three points are massive for us,” said Solskjaer.
“We’ve had too many draws this season and given too many points away from winning positions. It’s a great lesson the last two games (Sheffield United and Aston Villa) and we came back in a great manner.”
Mourinho, sacked by United last December, described his reception at Old Trafford as “nice and polite”.
“It’s a step back against a team that is playing with the same objective as us, trying to get up the table and reach the top six,” he said. “We need to keep going. We made mistakes but it’s not good to be crying about them now.”
Liverpool surged eight points clear after their comprehensive victory over struggling Everton at Anfield.
All but one of the goals came before halftime as Divock Origi and Xherdan Shaqiri, who replaced the rested Roberto Firmino and Mohamed Salah, put Liverpool 2-0 up inside 17 minutes.
Michael Keane pulled a goal back for Everton before Origi got his second and Sadio Mane finished off a blistering counter-attack to make it 4-1.
The scoring did not stop there as Richarlison cut the deficit once more, but Georginio Wijnaldum hammered the final nail in the Everton coffin as an eighth defeat in 11 league games sees the Toffees slip into the bottom three.
Leicester moved back above Manchester City as Liverpool’s closest challengers.
The Foxes were awarded a penalty after the break when Jonny Evans went down clutching his face inside the area and Jamie Vardy made no mistake, scoring for the seventh straight game.
James Maddison sealed the points deep into stoppage time.
Chelsea badly missed the presence of Tammy Abraham due to a hip injury in Saturday’s shock 1-0 home defeat to West Ham and the return of their top scorer proved the difference against Villa.
Abraham’s 26 goals on loan helped Villa get promoted last season and he did not celebrate when heading in Reece James’s cross to open the scoring.
Trezeguet bundled home to bring the visitors level before the break, but Abraham then turned provider by chesting down for Mason Mount to brilliant volley high past Tom Heaton.
Wolves, in fifth place, remain unbeaten in the league since September and goals from Leander Dendoncker and Patrick Cutrone either side of halftime gave them a 2-0 win against a disappointing West Ham at Molineux.
Southampton moved out of the bottom three with a 2-1 win against Norwich/AFP.