The Saints had conceded 11 goals in their previous four Premier League matches heading into their game at Old Trafford.
But United failed to find a way to break down Southampton’s rigid defence with Romelu Lukaku going off on a stretcher in the first half after a clash of heads.
The Belgium international has now been ruled out of United’s next two matches while Zlatan Ibrahimovic will be out of action for a month.
It leaves Mourinho short of attacking options going into the New Year with Marcus Rashford expected to come in to lead the line.
United also slipped down to third after the draw against Southampton with Liverpool, Tottenham and Arsenal also hot on their heels in the chase for a top four spot.
And Pogba has urged his side to return to winning ways to ease the pressure.
“When we have Romelu, he makes it difficult for defenders and creates so much space as well. He’s a big miss for us but we have the team to fill the gap of Romelu,” he told MUTV.
“We didn’t lose [against Southampton] but we have to wake up, bounce back and get back to winning.
“We drew again. Three draws we have to win, something has to change and we have to change to win.
“It’s only one day and then we have to play against Everton – that’s football. Tomorrow’s another day and we have to go out there and win.”