The Reds remain tied on points with United after the defeat. However, Ole Gunnar Solskjaer's men can overtake them when they visit Burnley on Jan. 12.
Southampton wasted no time going ahead at St. Mary's, with former Liverpool forward Danny Ings stinging his former team with a wonderful lobbed finish after being picked out behind a makeshift Liverpool defence by a James Ward-Prowse free kick in the second minute.
The goal marked the 50th Premier League goal of Ings' career between stints at Southampton (36), Burnley (11), and Liverpool (3).
Liverpool, who Jurgen Klopp said had been making "lemonade" out of a recent string of poor performances, could not recover a result this time around, setting the stage for United to take control of the league next week.
Southampton coach Ralph Hasenhuettl fell to his knees at the final whistle as his players celebrated.
"He [Hasenhuettl] is very passionate and he brings that passion into the game, and it's great for us to see that," goalscorer Ings told Sky Sports.
Klopp, whose side were held to a goalless draw by Newcastle United in their last outing, was disappointed with the display.
"They [Southampton] put a lot of work in. Our decision-making was just not good, that is how it is when you don't have momentum. We should have had much more chances," he said.
"The best situations we had were with Sadio having the ball at his feet," Klopp told Sky Sports. "For anyone to say he's a diver is the biggest joke ... He had two situations where other teams would get a penalty.
"The handball, I don't know who will explain that to me. In these situations we were unfortunate ... I hear Manchester United had more penalties in two years than I had in five-and-a-half years. I have no idea if that's my fault, or how that can happen.
"But it's no excuse for the performance. We cannot change it, we have to respect the decisions. We can change our performance. That's our focus now."
After an FA Cup third-round match against Aston Villa on Friday, Liverpool's next match in the Premier League will be at home against Man United on Jan. 17.