The result maintains United's unbeaten start to their league campaign, although they remain second in the table, behind Manchester City on goal difference.
Here is how Sports Mole covered the action at St Mary's Stadium.
2.00pmHello and welcome to Sports Mole's live coverage of today's Premier League clash between Southampton and Manchester United.
2.03pmToday's encounter at St Mary's sees a Saints side that has made a decent enough start to the campaign up against a team that has hit the ground running in impressive fashion, sitting second in the table with 16 goals from their first five games.
2.05pmOne of the key questions today is whether Virgil van Dijk will start for Southampton against one of the strongest attacks in the league at present. The team news are out so let's find out.
2.08pmMANCHESTER UNITED STARTING XI: De Gea, Valencia, Jones, Bailly, Young, Fellaini, Matic, Mata, Mkhitaryan, Rashford, Lukaku
2.09pmSOUTHAMPTON SUBS: McCarthy, Austin, Ward-Prowse, Van Dijk, Gabbiadini, Hojbjerg, Pied
2.10pmMAN UNITED SUBS: Romero, Smalling, Carrick, Blind, Herrera, Lingard, Martial
2.12pmSo the answer to the Van Dijk question is no. The Saints defender, working his way back to full fitness, has to make do with a place on the bench. Southampton are otherwise unchanged following their 1-0 win at Crystal Palace last Saturday.
2.14pmAs for Man United, Jose Mourinho has kept faith in the side that claimed an impressive 4-0 win over Everton last Sunday, with no changes in the starting XI.
2.16pmThis means that Marouane Fellaini starts in United's midfield for the second week running, and also that Juan Mata and Marcus Rashford will have started three games in a week after also playing in the EFL Cup win over Burton Albion on Wednesday.
2.18pmDaley Blind replaces Matteo Darmian on the bench, but there is no place in the squad for left-back Luke Shaw against his former side, Mourinho having told the press that the England man needs to improve if he is to return to contention.
2.20pmDespite Saints' reasonable position in the table, sitting ninth with just one defeat in five, they have been struggling to find goals at home. Just one home goal from free play in 725 minutes, in fact, with eight home scoring blanks in nine matches.
2.22pmOn the plus side, Saints they have kept three clean sheets in five games this season, but their defence - still without Van Dijk - will be tested to the limit this afternoon against a free-scoring Man United side.
2.25pmA team with an average of 3.2 goals a game in the five league matches so far this season, plus another four in their win over Burton in the EFL and three in the Champions League against Basel, is what awaits the Southampton back line today.
2.27pmAnd very much harking back to the Fergie days, the current United crop knows how to find late goals - the Red Devils have scored nine times in the 80th minute or later this season.
2.30pmSo what have the managers of their respective sides said about today's encounter? Let's begin with Saints boss Mauricio Pellegrino.
2.32pmThe Argentine was full of praise for the opposition, telling reporters: "When we play against the biggest team in the world, it allows us to draw some conclusions about how we are. This is for me the nice challenge and we have to strive for this. Playing against Manchester United is one of the most beautiful things. I will live this opportunity, I will live this game, and now we have to try to enjoy this game."
2.35pmJose Mourinho, ever the pragmatist, told reporters in Friday's press conference: "Southampton are the kind of club that are very stable in the way they play and in the way they see football, the qualities of their players... I see the match tomorrow as a very difficult match."
2.37pmOne key player who will be absent for United is Paul Pogba, out for the third game running with a hamstring injury, and Mourinho has not given a timescale for the 24-year-old midfielder's return.
2.39pm"I have no idea," Mourinho replied at his press conference when asked about Pogba's return date. "No idea. But, again, it's a great opportunity for other players to play and I trust them all. So I'm not going to cry or be counting the days until Pogba comes back, not at all. When he comes back, welcome, but until then, I trust my people."
2.41pmHEAD TO HEAD: The statistics surrounding this fixture make for some bad reading for Southampton fans, I'm afraid.
2.42pmSouthampton and Man United have met 121 times in competitive fixtures. Saints have won 28 of those, 30 have been draws, and 63 are United wins.
2.44pmSouthampton have lost 24 Premier League games against Manchester United, which is their worst record against any opponent - four more defeats than they have suffered against anyone else.
2.46pmMan United have also won 16 of the last 22 meetings against Saints, losing just three.
2.48pmAnd when it comes to form at St Mary's, United have won 14 of a possible 18 points in their last six Premier League visits to Southampton's ground.
2.51pmPREDICTION: Despite all the stats surrounding the fixture heavily stacked in favour of Man United, I see this game as being tricky for the visitors today. Southampton have several good players and their defence has been on form, so they should give Mourinho's side a tough time. That said, I can see United breaking the opposition down with another late goal, then maybe adding another one right at the death, and leave the South Coast with a 1-0 or 2-0 win.
2.57pmBoth sets of players are out onto the St Mary's pitch, with kickoff just moments away!
0 minKICKOFF! Craig Pawson is the referee for today's match and he gets the action underway!
2 minAn early stoppage as Tadic accidentally kicks Young in the chest as the Saints man tried to latch onto the ball. Young appears winded and in pain but eventually manages to get back up.
4 minThe ball is lumped up to Long in a dangerous position, but he can't make anything of it as Jones clears.
6 minRedmond does well down the left as he sends a cross in, but Jones clears ahead of Long, with the ball then falling to Rashford, but Cedric hacks the youngster down to snuff out the counter.
8 minCHANCE! We have the first effort on target and it comes courtesy of Mkhitaryan, who picks up a loose ball outside the Saints box, but Forster has it covered and gets down low to save.
11 minCHANCE! Young miscues a clearance to gift Southampton a corner, it's taken and another poor ball out from Fellaini sees Redmond have a go - he strikes it on the volley 15 yards out and it just flies wide!
14 minNow it's United who win a corner, which is floated towards Fellaini, the Belgian then goes down alongside Romeu, but the free kick goes Southampton's way.
16 minJust over quarter of an hour played and it remains goalless, neither side has really gotten into gear yet but Saints have enjoyed more of the possession.
20 minGOAL! SOUTHAMPTON 0-1 MANCHESTER UNITED (ROMELU LUKAKU)
21 minWe have the breakthrough at St Mary's as United take the lead, and who else but Lukaku to get his sixth of the league campaign! Young crosses into the box for the Belgian to head goalwards, Forster does well to keep the ball out but the rebound falls to Lukaku who duly pokes home!
25 minSouthampton are exchanging some short passes to try and open up the opposition and Cedric tries to send a cross into the box, but it's cleared. Redmond then tries to get onto the rebound but his effort is charged down.
27 minCHANCE! Mata is upended by Yoshida to gift United a free kick in a good position, Rashford takes and curls the ball over the wall but it's inches wide of goal.
30 minHalf an hour played and United have the lead as well as edging ahead in terms of possession, but Saints currently have the ball and are trying to find gaps in the opposition defence, though United are remaining tight and not letting anything pass.
32 minMkhitaryan plays a one-two with Rashford and tries to reclaim the ball in the area, only for Lemina to barge the Armenian off the ball. Nothing given though and no complaints from United.
36 minSome more tentative forays by United as Fellaini sends a long ball to Lukaku, who tries to nod it towards Mkhitaryan, but Saints clear, and not long after Mkhitaryan embarks on a run and forces a corner but Mata's ball into Bailly ends up into the arms of Forster.
38 minBOOKING: Long drags his studs into the back of Fellaini's heel, sending the Belgian to ground, and the Saints man is shown a yellow card for his troubles.
40 minFellaini is back on after some treatment but isn't moving too freely.
43 minMata send a low pass into the box that is blocked by the one of the defenders and goes out for a corner, which the Spaniard opts to take himself, but it's easily cleared.
44 minBOOKING: Young is cautioned after clattering into the back of Tadic.
45 minLukaku's low cross is diverted away by Hoedt as the officials announce two minutes of stoppage time.
45+2 minHALF TIME: SOUTHAMPTON 0-1 MANCHESTER UNITED
3.48pmThat's half time here at St Mary's and Manchester United are a goal up against Southampton, Romelu Lukaku continuing his rich vein of form with his sixth goal of the league campaign and eighth in all competitions.
3.52pmThe Belgian put his side ahead in the 20th minute after some good work from Ashley Young, who send a pinpoint cross into the box for the former Evertonian to head goalwards. Fraser Forster did very well to keep out the header, but the ball fell straight back to Lukaku, who had the easy task of rifling home from close range.
3.55pmSaints have done well to keep United at bay and actually enjoyed narrowly more possession - 51% to 49% - but are yet to really test David de Gea, so whether they will be able to get back into this remains to be seen.
3.58pmAt the top of the Premier League table it remains all even in terms of standings, with Manchester City leading 1-0 at Crystal Palace at the break. Will one of the two Manchester clubs edge ahead on points or goals come the final whistle? The next 45 minutes are coming up shortly.
46 minKICKOFF: Referee Craig Pawson gets the second half underway.
47 minSaints with the better of the early exchanges as Lemina leads a break and Tadic wins a corner on the right.
48 minDavis swings the ball in but it takes a deflection and United clear their lines.
51 minEnd-to-end stuff as Long tries to get onto the end of a Bertrand ball, but Bailly clears, and United break at the other end only for Hoedt to intercept the ball and get it away.
53 minCHANCE! Saints win a corner which Tadic takes, Yoshida flicks on, the ball then comes off Bailly and falls to Romeu, but he fires wide under pressure from Jones!
55 minAdvantage Man City at the top of the table...
GOAL Man City 2-0 Crystal Palace (51 mins)
Raheem Sterling taps in from close range after a perfect cross by Leroy Sane #MCICRY
57 minCHANCE! Saints are working up the gears and test De Gea through Redmond, who cuts inside on the edge of the box and aims for the near corner of the net, but the United keeper is able to claim at the second attempt.
60 minWe're an hour in and Southampton win yet another corner, which Davis takes but is eventually cleared to safety. Mourinho is not looking happy on the touchline.
63 minPENALTY SHOUT! Long goes down under the challenge of Jones right on the edge of the box, cue calls to award a spot kick from Saints, but Pawson waves play on!
65 minUnited have very much relinquished what control they had in this game but they muster an attack now - their first of the second half - as Lukaku threads the ball into the path of Mkhitaryan, but the latter's effort is deflected and loops into Forster's arms.
67 minCHANCE! Southampton are caught on the counter, with Mkhitaryan sliding a pass through to Lukaku, who shoots low and hard towards goal, but Forster keeps the effort out with his feet!
68 minOFF THE LINE! But here come Saints as they win a corner, Romeu gets a header towards goal, De Gea is beaten, but Fellaini is in the right spot to head off the line!
70 minCHANCE! Man United on the receiving end of yet more attacking play as Redmond decides to have a go from 25 yards out, but De Gea gets his body behind the ball and catches the effort with ease.
70 minBOOKING: Valencia clatters into Davis and earns a yellow card for his troubles.
72 minSouthampton remain on top as Soares whips a ball in from the right, which Bailly clears, only for Tadic to have another go with a delivery which De Gea manages to pluck out of the air.
74 minSUBSTITUTIONS: One apiece as Davis makes way for Manolo Gabbiadini in the Saints XI, while Chris Smalling replaces Mkhitaryan for United.
78 minCHANCE! So close to an equaliser for Southampton as Tadic slips the ball into the path of Romeu at an angle, the Spaniard tries to find Gabbiadini but it's just too heavy for him as the ball rolls narrowly wide of the far post!
81 minCHANCE! But here come United, with substitute Herrera put clean through on goal by Lukaku, but the Spaniard gets right under the ball and sends it just over the bar!
82 minBOOKING: Lukaku is booked for clattering into Lemina.
84 minSUBSTITUTIONS: A double change from the Saints as Long and Cedric come off, and Charlie Austin and James Ward-Prowse come on.
88 minBOOKING: Matic is shown a yellow card after bringing down Tadic.
90 minFour minutes of added time for United to hang on here at St Mary's.
90+3 minSUBSTITUTION: Rashford makes way for Daley Blind for the final minute of the match.
90+4 minJose Mourinho has been sent to the stands! More on that as we get it...
90+5 minFULL TIME: SOUTHAMPTON 0-1 MANCHESTER UNITED
4.53pmAnd that's it! Man United hang on for all three points as they claim a 1-0 win over Southampton at St Mary's Stadium.
4.55pmRomelu Lukaku's strike in the 20th minute turned out to be the winner for the visitors, while the hosts should get credit for a much-improved second-half display, but United's defence held firm. Mourinho was then sent off late on for venturing onto the pitch in the final moments.
4.56pmThat's all we have time for but don't go anywhere because Sports Mole will be bringing you match reports, reaction and all the latest news from the day's football. Thank you for following and, until next time, goodbye!