Manchester United again have Anthony Martial to thank for victory, as the teenager showed impressive calmness in front of goal.
United fell behind to a Graziano Pelle goal in the first half, but Martial seized on a loose ball in the box, pulled off a neat turn to leave Virgil van Dijk's brain on the floor, and then fired past Martin Stekelenburg with a cool finish.
In the second half, Antonio Valencia replaced Matteo Darmian who endured a relatively poor half along with most of the United defence, and had Maya Yoshida to thank when his backpass was latched onto by Martial, who again was unflappable in front of the 'keeper.
United made it three when Memphis' quick turn in the box made room for him to slam a shot against the post. The ball ricocheted into the path of Juan Mata,who coolly finished.
Louis van Gaal then introduced Paddy McNair for Marcos Rojo, only to see Pelle split him and Smalling in the box for a simple header to half the deficit for Southampton. United held on, and were not panicked as they did so, but David de Gea pulled off a couple of excellent saves, notably one at full stretch. United have now moved into second position, just two points behind Manchester City.