United failed to capitalise on a missed kick from Alex McCarthy three minutes in, and they were punished when Stuart Armstrong blasted across David de Gea's goal and into the net with Saints' first effort on target.
Rashford's profligacy in front of goal against Young Boys in the Champions League in midweek provoked Mourinho to display his frustration on the sidelines.
David de Gea was forced into a smart late save to preserve a point for United, who are now six points behind fourth-placed Chelsea having played a game more, while Southampton stayed in the bottom three and remained winless at home.
The Portuguese manager accused the 25-year-old of showing a lack of respect towards his fellow players and the Man United fans. "Doesn't matter the system we play, has to do with the characteristics of the players". The ones that bite the ball all the time and press all the time, we don't have many people with that spirit so nothing to do with the system.
Saturday's game with Southampton certainly looked like a relatively simple way to get positive momentum going but it proved to be anything but.
United lined up with midfielders Scott McTominay and Nemanja Matic alongside Phil Jones in what was a reshuffled defence, despite Marcos Rojo being on the bench, and when asked by BT Sport who will be available for the game against Arsenal on Wednesday, Mourinho replied: "I don't know". I would say the only problem that we had was that the two attacking players didn't have enough continuity because in midfield we lost too many balls.
His performance on Tuesday night was a very good one given the circumstances and I expect him to come in for the injured Lindelof. Great examples of players fighting until the limits.
But Mourinho feels the time when they could take their pick of the standout talents at domestic rivals is over.