The Belgian striker scored the game's only goal on 20 minutes, firing in the rebound after his header had been kept out by Saints goalkeeper Fraser Forster.
The hosts started to come into their own at the beginning of the second period with chances for Oriol Romeu and Nathan Redmond but couldn't find the equaliser.
The game began to pick up as Lukaku came close to a second before Marouane Fellaini's crucial clearance ensured that all three points would be heading back to the North West.
Continuing his fine start
The two Manchester clubs have proved the early pace setters in the top-flight and with both playing at 3pm on Saturday it was crucial that José Mourinho's side maintained their good start.
They faced a tough trip to the South Coast against Mauricio Pellegrino's men but Lukaku continued his excellent start to his United career with the only goal of the clash after 20 minutes.
The Belgian had Ashley Young to thank for his eighth goal of the campaign as the winger did brilliantly down the left to twist and turn before firing in a cross.
Lukaku's initial header was brilliantly saved by Forster but the rebound fell straight back into the striker to fire home at the second time of asking.
Marcus Rashford was inches away from making it 2-0 seven minutes later when the youngster guided a free-kick that went straight over the wall and had Forster well and truly beaten, but flew just a whisker wide.
Starting to come into their own
The Saints had hardly been in the clash in the first 45 minutes but perked up at the beginning of the second, and they had two good chances to find an equaliser.
The first came in the 53rd minute as Dušan Tadić's corner was flicked on by Maya Yoshida. The touch from the Japanese international fell to Romeu with help from an Eric Bailly deflection, but the midfielder diverted his effort wide under pressure from Phil Jones.
Redmond had a bit more luck minutes later as he managed to cut inside into the area and shoot towards the near post, but David De Gea was down well to save.
Coming very close
As the second period went on the chances began to dry up slightly but as far as Mourinho was concerned United had done their job. Yet a hectic two minutes nearly changed the tide of the clash.
United should of been 2-0 up in the 67th minute as Henrikh Mkhitaryan picked up the ball high up the field. He played it to Lukaku on the right who decided to go for goal at the angle and Forster did well to save with his feet.
Seconds later it was Southampton's turn to go close as Romeu did brilliantly to wrestle away and get his head onto a corner. The Spaniard looked to have collected a point but Fellaini did brilliantly to clear off the line and protect United's advantage.
The points were all stitched up as the game entered extra-time but it didn't mean that there couldn't be late controversy as Mourinho was sent off and forced to go to the stands.
Referee Craig Pawson gave the United manager his marching orders after he stepped onto the field of play, but the decision had no impact on proceedings as the Red Devils kept pace with City at the top of the table - behind only on goal difference.