Goals from Zlatan Ibrahimovic and Jesse Lingard gave United a 2-0 lead against the run of play.
But Manolo Gabbiadini scored either side of half-time to deservedly draw the Saints level.
However Ibrahimovic got his 26th goal of the season with a late header.
And after the match Mourinho was gracious enough to praise Southampton for their effort and acknowledged the problems his side had.
He told MUTV: “For us it was hard and difficult.
“Easy to feel the difference in freshness. They were fresher, sharper and quicker.
“We had difficulties in midfield. Big pitch, fast pitch, the game broken, it was difficult.
“The game came to our side but could have gone to their side. I think they played extremely well.”
