It had started well for United with a goal after just two minutes from Zlatan Ibrahimovic following a flowing move.
United beat West Ham 4-1 at Old Trafford, three days after drawing 1-1 with the same team at home.
The news is a blow to manager Slaven Bilic, whose side meet United again in the EFL Cup on Wednesday before a league game against Arsenal on Saturday.
"But in the second half the team again got the confidence to play and to win comfortably and playing lovely, attacking football, which is something the fans like, but we need results".
"That's how we have played [all season]".
Among the other positives for United were the excellence of Henrikh Mkhitaryan, continuing his recent rejuvenation, and the return from exile of Bastian Schweinsteiger with a late cameo off the bench.
"I'm so happy because today was against a Premier League team with Premier League qualities with Premier League characteristics, ' Mourinho told Sky Sports after the game".
"One of the things I proposed to myself was to come to such a club like Manchester United who play the kind of football that people want, because it's Manchester United's tradition, so I'm really happy with that".
However, United were able to re-establish their hold on the game and clinch a place in the last four.
"I enjoy this win for all of the squad and for the fans, because it's a good answer to keep all of our possibilities in every competition", said Southampton manager Claude Puel.
West Ham's cause was not helped by the fact they lost both wing-backs, Aaron Cresswell and Michail Antonio to injury, and in the 62nd minute they found themselves 3-1 down. Forget that at half-time we should be winning and we aren't.