Seven months on from edging an entertaining EFL Cup final against Saints, United arrived at St Mary's looking to maintain their fine start to the campaign against a team struggling for coherence and goals on home turf.
United were made to sweat as Mauricio Pellegrino's men launched wave after wave of attacks in the second period, but Lukaku's first-half goal was enough to secure a 1-0 win, taking his side to a club record-equalling 16 points at this stage of a Premier League season.
United are now 11/4 with Sky Bet to win the Premier League title.
However, while it may have been a successful trip on the field, things did not go so well off it. Mourinho was sent off in stoppage time by referee Craig Pawson for his antics on the touchline, while in the first half some supporters went against United and Lukaku's plea to stop chanting a song about the striker that Kick It Out have called "offensive and discriminatory".
United are understood to be preparing a strong response after the song echoed around the ground following the Belgian striker's eighth goal in as many games for his club, having tucked home after his initial header was denied by Fraser Forster.
Post-match reaction
Jose Mourinho insisted he did not know why he was sent off in the final minutes of Manchester United's victory as he bizarrely blamed their underwhelming performance on the good weather.
Asked why he had been sent off, the Portuguese responded: "I don't know. Craig told me to leave, and I left.
"Did I ask for an explanation? No. He told me to leave, and I left.
"(I collided) with the fourth official (Mike Jones).
"(My players) were really tired at the end. Nine of them didn't play in midweek (in the 4-1 defeat of Burton) so nine of them, it's not even an accumulation of fatigue.
"The weather was beautiful. Even if you train at 11am, 12pm, 3pm in Manchester, we don't get this weather. Seriously. The last time we had this was in Los Angeles and then in Macedonia, and they felt (it).
"I saw some players not sharp. Great spirit and always trying, but not the same sharpness. So if your opponent is coming with everything against you, let's do it for 10 or 15 minutes and stay solid.
"We are better this season than last. But this is nothing. It's just a start. I really think that six teams are going to fight for the title."
Mauricio Pellegrino's Southampton failed to score for the fifth time in seven fixtures since his summer appointment as manager and he said: "You have to score goals.
"I don't like to talk about good luck or bad luck, because we have to score goals and we couldn't.
"But, at the same time, we played a good game. In general we controlled the game against United, a team who are really difficult to control.
"To play in this way with this spirit, with this football, we'll win more games than we lose. I am sure. This is the best thing for us. We have to continue in this way."
