Chances were scarce at both ends, but the visitors spent the whole of the second half on the back foot, seemingly content to allow Southampton the ball and maintain a strong defensive shape.
Mourinho, who was sent to the stands in the dying minutes by referee Craig Pawson after he was summoned by the fourth official, says the result was all that mattered.
"It is not always possible to operate at high quality, but we fought hard," the United manager told Sky Sports. "We did what many teams in the Premier League do for 90 minutes, which was play with five defenders at the back.
"Credit to Southampton: they tried to play. [Mauricio] Pellegrino changed the team and sent on another striker to play more direct. It was a hard game, but I am really happy.
"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."
He added to MUTV: "Three fantastic points – the same points as when we win 4-0.
"Our team had chances to kill the game. We had two enormous chances with [Ander] Herrera and Lukaku. Then we put ourselves in a position where he had to defend to keep the result, which we did extremely well, with a good keeper and with good defenders, so I am really happy."
Southampton striker Shane Long felt his side were worthy of a point, despite them managing only four reasonably tame efforts on target throughout the 90 minutes.
"I thought we deserved more from the game," he told Sky Sports.
"Manchester United are dangerous all over the pitch. They have physical players on set-pieces. To limit them to one goal was a credit to our team, but it was just a shame we couldn't get that equaliser at the other end.
"There were a lot of chances, but things didn't seem to fall for us. It wasn't through a lack of trying though. It just didn't happen."