The result leaves the champions in eighth spot, eight points behind leaders Arsenal after eight games played.
Saints are four points better off than United at this stage; they slip out of the top four to fifth place.
Mauricio Pochettino's side started the game with energy and a high intensity and could have taken the lead early in the first half.
Jay Rodriguez sent a warning to the home side when twice heading wide and Dani Osvaldo really should have done better when the ball dropped to him on the edge of the area after Victor Wanyama robbed Wayne Rooney of possession deep inside the United half.
It was the home side who took the lead though when Robin van Persie scored after 26 minutes - United's first goal from open play at home in the Premier League this season.
It was a slow start from the champions but Adnan Januzaj's sublime pass to Rooney provided the England forward with the chance to score. Artur Boruc smothered well but the danger was not cleared and Van Persie converted excellently from a tight angle.
The goal appeared to give the home side and the fans a lift.
Moments after Van Persie's opener, Nani found space on the right and his cut-back to Rooney cannoned off the crossbar.
After the break, United went in search of that killer second but Van Persie could only flick Rooney's expertly-taken corner onto the bar.
Man-of-the-moment Januzaj then tested Boruc from distance but the Polish keeper stood firm and his solid arm diverted the ball to safety.
Southampton then regained control of the ball and the game.
Osvaldo again failed to hit the target from the edge of the area and, when he eventually got the better of Phil Jones in the air, he could not direct his header at goal.
But, when it looked like United were going to hang on for all three points, Dejan Lovren caused mayhem in the United penalty area and his effort cannoned off Lallana and into the empty net.
The goal was a fitting reward for the travelling fans but for United, further questions will be asked of Moyes after more vital points are dropped at home.
The Red Devils host Real Sociedad in the Champions League on Wednesday before a home game against Stoke next Saturday, when Saints entertain Fulham in the late kick-off.