A solitary effort from Romelu Lukaku early in the contest was all that separated the two sides on the day, with Mourinho's men forced to fend off an increasingly-potent Saints attack in the closing stages.
United had scored at least twice in every league game this season prior to today and speaking afterwards, Mourinho admitted that his side would not always play with a high-intensity attack.
"It is not always possible to operate at high quality but fought hard," he told reporters. "We did what many teams in the Premier League do for 90 minutes, which was play with five defenders at the back.
"Credit to them, they tried to play. [Mauricio] Pellegrino changed the team and sent on a second striker to play more direct. It was a hard game but we fought really hard and I am really happy.
"Nobody feels the pressure to play always attacking football because so many teams play always defensively. I felt some of the boys were not sharp. I didn't feel like we could score three or four goals like we sometimes do so the option was to make sure we got the points."
Next up for United, who remain level on points with Manchester City at the top of the table, is a midweek trip to CSKA Moscow in the Champions League ahead of a home encounter with rock-bottom Crystal Palace next weekend.