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Liverpool were left to rue a slow start as Rashford twice finished with astonishing verve to lift second-placed United five points clear of the visitors after Romelu Lukaku had outmuscled Dejan Lovren.
The visitors dominated after the break, but could only find a response via a United player when Eric Bailly turned into his own net 24 minutes from time.
“If people don’t think we deserved it, I don’t care,” Mourinho said.
“We can split the halves. United first half and Liverpool second but in my opinion our first half was a half with goals and danger.
“In the second half… Was Liverpool controlling with the ball and United controlling without the ball.”
Just a second defeat in 21 Premier League games leaves Liverpool still in third, but they could drop to fourth if Tottenham win at Bournemouth on Sunday.
And Liverpool manager Jurgen Klopp raged at the decision not to award his side a penalty when Sadio Mane tumbled under a challenge from Marouane Felliani as Klopp’s men saw three penalty appeals in all waved away.
“Second half we had to continue chasing the game, then we scored the goal and should have been a penalty around Felliani situation on Sadio,” said Klopp. “It was best piece of football in the whole game.”
And Klopp claimed his job would have been made “massively” easier had penalty decisions gone in his side’s favour in recent weeks after conceding a stoppage time spot-kick to draw 2-2 with Tottenham last month.
“If the assistant against Tottenham didn’t see, which can happen, we win 2-1 against Tottenham, and today if he would see it then it’s a draw.
“We can count how many points that would be more and it would change my life massively.”
– Rashford ‘fundamental’ –
Rashford may not have even have started but for an injury to Paul Pogba suffered in training on Friday.
Without the £89 million ($123 million) French international, Mourinho moved Alexis Sanchez inside behind Lukaku and handed Rashford his first league start of the year on the left.
“He scored two goals in a big match against a top class opponent,” added Mourinho. “Of course he was fundamental.”
The Portuguese coach’s plans worked to perfection when Rashford brought the game to life with a superb finish after 14 minutes.
Lukaku beat Lovren to a long ball and the England international skipped inside Trent Alexander-Arnold before smashing a shot past the helpless Loris Karius.
Soon Liverpool’s task became doubly difficult as United’s power and pace again proved too much.
Lukaku shrugged off Lovren once more and when his ball through to Juan Mata was cut out by Van Dijk, Rashford swept home the rebound.
“Lukaku was confident by his experience that he could be dominant in relation to Lovren and it worked,” said Mourinho.
An eventful afternoon for Rashford continued as he picked up a yellow card and then could easily have added an assist to his double when Mata pulled an acrobatic effort eide.
Liverpool enjoyed almost complete control of the ball in the second half but failed to find a way through the mass ranks of United defence.
When Liverpool were handed a lifeline, it came via a United boot as Bailly — who had excelled on his first start since November — flicked Mane’s cross into his own net.
Mourinho then drew the ire of his own fans as he replaced Rashford with Marouane Felliani 20 minutes from time to see the game out.
And that change could have backfired had referee Martin Atkinson not dismissed claims when Mane went down under a challenge by the Belgian to Klopp’s despair.
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