Rashford beat Tottenham keeper Paulo Gazzaniga at his near post after six minutes after the ball had broken to the England forward off Davinson Sanchez.
Then, after a long wait for a VAR check, Rashford kept his nerve to convert a penalty four minutes after the break, once it had been ruled the striker had been fouled by Moussa Sissoko.
Dele Alli had equalised with a brilliant goal at the end of the first half but United were good value for their victory after creating a number of excellent chances they failed to take.
Rashford was unable to become the first United player to score a league hat-trick since Robin van Persie’s memorable effort against Aston Villa in 2013 but he now has 12 goals in 13 games for club and country, and his nine Premier League goals leave him one short of his season best.
Mourinho’s return well-received
As expected, Mourinho was well received by the United fans, who never fell out with their former manager and have no particular axe to grind with him.
That respect will never match the affection Old Trafford has for Solskjaer though.
And the Norwegian used memories from his playing days to get the crowd up for the game by emerging last from the tunnel, triggering a song in his honour and the start of what proved to be a lively atmosphere.
Evidence of change at United came with a team that contained only five players Mourinho picked for the corresponding fixture last season.
That August night ended in a 3-0 defeat for United and an angry Mourinho demand for the media to show him some “respect” for his three Premier League titles.
The first of those triumphs is over 15 years ago now. Mourinho’s task is to show his best days are not behind him.
He didn’t make a particularly brilliant job of that in his last weeks in Manchester and, as happened so often then, tonight he spent long periods in his technical area with his hands in his pockets watching his team get outplayed.
His substitutes failed to inspire and with eight goals conceded in four games, Mourinho evidently has some work to do defensively.
At the end, he moved to shake Solskjaer’s hand before striding purposefully away to try and lift his players.
Dele’s delightful goal
For months, there had been a debate about what had happened to Dele Alli.
Once one of the golden boys of the English game, he had been reduced in influence and effectiveness and lost his place in Gareth Southgate’s national squad.
Who knew the answer was replacing the manager he loved?
One of the first things Mourinho did after replacing Mauricio Pochettino was to ask Alli whether it was him or his brother who had been playing for Tottenham in recent times.
This is definitely him.
His third goal in three Premier League games – he only scored three in his last 17 under Pochettino – was extraordinary.
Fred thought he had the situation under control as the ball looped up on the edge of the six-yard box.
But Alli leaned into the Brazilian, then rolled round him after a beautiful piece of control before turning a shot past De Gea into the far corner.
It was as breathtaking as the Cristiano Ronaldo-esque 35-yard shot Rashford rattled the bar with – and Mourinho loved it.
In his pre-match press conference, Solskjaer had dismissed as “lies” suggestions he had told his players he would be sacked if United lost against Tottenham, and again at Manchester City on Saturday.
The word remains from United that the Norwegian is under no immediate danger of losing his job amid an acceptance from those in senior positions that there will be bumps in the road this season.
This was the type of performance that gives credence to Solskjaer’s belief genuine progress is being made.
Yet one look at the respective substitutes’ bench shows United are crying out for reinforcements when the transfer window opens next month.
Whereas Mourinho had six experienced full internationals to turn to as he tried to change the game in Tottenham’s favour, Solskjaer had two and neither Luke Shaw nor Juan Mata have won a cap for quite some time.
At the end, Solskjaer milked the rapturous reception he was given.
If Sheffield United and Arsenal fail to win on Thursday, United will stay sixth. In order to stay there, Solskjaer will need more than the crowd behind him.