United had a great chance to score just three minutes in after Alex McCarthy stumbled when trying to clear the ball, but recovered in time to deny Lukaku.
Lukaku scored his first goal since mid-September when he burst onto a deft Rashford touch pass to make it 2-1 in the 34th.
Still seventh in the table, the Red Devils could end the weekend nine points off the top four.
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Romelu Lukaku and Ander Herrera restored parity before the break, but United were unable to find a victor during an underwhelming second half performance during which Pogba repeatedly conceded possession, much to the visible frustration of his manager.
Mourinho made five changes to the side that beat Young Boys in the Champions League, with injuries forcing the Portuguese to deploy three central defenders. Can we go there and bring those players here? No.
"But our two attacking players didn't have support because we kept losing the ball in midfield".
"We don't have many mad dogs, the ones who bite all the ball all the team and press all the time, we don't have many with that spirit", he said.
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The list is quite long to mention all the teams that have managed to put at least two goals past De Gea, but it is especially worrying when that does a team like Southampton.
The duo can perhaps be excused to a certain degree, given they were playing out of position, but this was not a system or formation that favoured United, as they lacked the requisite intensity to really thwart the Saints, as well as the requisite resolution to stifle their attack.
Rashford was the provider again as United equalised after 39 minutes, running into the box and teeing up Herrera to finish from close range.
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Nathan Redmond had one of his best performances this season come against Manchester United.