Romelu Lukaku scored his sixth goal of the season, beating Fraser Forster at the second chance after the Southampton keeper made a brilliant save on the initial header.
Oriol Romeu and Shane Long each had an opportunity for a late equaliser, but the duo matched Southampton’s woes in front of goal.
Southampton has scored just four times and taken eight points from their opening six Premier League games.
The hosts made a competitive start, but saw their work cancelled out as Lukaku gave United a 1-0 lead after 20 minutes.
Lukaku was left free in the box to put Ashley Young’s cross on target, but Forster was equal to the task. The rebound, however, fell right at the striker’s feet and he smashed home into a gaping net with the keeper on the deck.
Marcus Rashford came close to adding a second just five minutes later when he fired wide of the upright from a 25-yard free kick, which had Forster beaten.
United continued to mount pressure on the Southampton goal and Wesley Hoedt did well to prevent Lukaku’s cross from finding a charging Rashford at the back post nearing half-time.
Romeu pounced on a loose ball in the box shortly after the restart, but Phil Jones was there to apply the pressure and the midfielder shot agonisingly wide of the goal.
Forster, whose form has picked up, denied Lukaku a second goal with a low save in the 66th-minute.
It would be Romeu again missing the target from close-range after great build up play saw Dusan Tadic play the Spaniard on goal. But the attempt across the net rolled helplessly by the post.
Ander Herrera missed an opportunity to put the game away when Lukaku found the player on the edge of the box, but too much time led to an attempt that cleared the bar.
Southampton pressed late too no avail and despite a spirited performance was left with another home defeat.
Southampton: Forster, Bertrand, Yoshida, Hoedt, Cedric (Ward-Prowse 82), Romeu, Lemina, Redmond, Davis (Gabbiadini 73), Tadic, Long (Austin 82).
Unused subs: McCarthy (GK), Pied, van Dijk, Hojbjerg.
Bookings: Long (37).
Manchester United: De Gea, Valencia, Bailly, Jones, Young, Matic, Fellaini, Mata (Ander Herrera 62), Mkhitaryan (Smalling 75), Rashford (Blind 92), Lukaku.
Unused subs: Romero S. (GK), Blind, Carrick, Lingard, Martial.
Goals: Lukaku (20).
Bookings: Young (43), Valencia (70), Lukaku (82), Matic (88).