Ole Gunnar Solskjaer's side looked to be heading for a fourth home game in a row without a win in all competitions after James Ward-Prowse converted a superb free-kick.
United, who lost Alexis Sanchez to injury, had overturned Yan Valery's opener through Andreas Pereira and Lukaku only to see the lead slip away.
But Lukaku's emphatic finish in the 88th minute was enough for all three points, even though Paul Pogba missed an injury-time penalty.
Lukaku was twice denied by Angus Gunn in the first five minutes, although United's early rhythm was disrupted by treatment on a Scott McTominay head injury.
Jannik Vestergaard headed over when picked out unmarked from a corner, and Ryan Bertrand was angry not to win a penalty from a shirt pull by Chris Smalling.
Saints' increasing pressure told 26 minutes in, when Valery's ferocious strike from 25 yards beat David de Gea at his left-hand post, even though he got hands to the ball.
United were struggling to impose themselves and suffered a further blow when Sanchez had to go off with a knee injury seven minutes after the restart.
However, they turned the scoreline around in a spectacular six-minute spell. First, Pereira curled a spectacular strike beyond Gunn from 25 yards out, before Lukaku got free in the box, cut onto his right foot and drilled a shot into the bottom-right corner.
10 - Romelu Lukaku has been involved in 10 goals in his last 11 Premier League appearances against Southampton (8 goals, 2 assists). Saintly.— OptaJoe (@OptaJoe) March 2, 2019
Just as United appeared to be heading for victory, Ward-Prowse sent an unstoppable effort beyond De Gea to send the travelling fans into raptures and silence the home fans.
But with two minutes of normal time remaining, Lukaku curled another low right-foot strike from outside the box past Gunn, prompting further cries of 'Ole, Ole, Ole' from the Stretford End, which were only slightly dampened when Gunn kicked away Pogba's penalty.
What does it mean? Lukaku restores home comforts
This was United's first home win since January 19 and marked the first time they have come from behind to win a match under Solskjaer.
Southampton stay 17th, two points above the drop zone.
Lukaku the main man again
Lukaku's performance followed a similar pattern to the 3-1 win at Crystal Palace. Having missed simpler early chances, he made the difference thanks to unerring accuracy with his weaker foot.
Young led a merry dance
Usually so dependable, Ashley Young looked vulnerable as Saints targeted the United right. He could have conceded a penalty for a tug on Ryan Bertrand and he gave away the free-kick from which Ward-Prowse scored. He was given little protection from midfield, though.
Key Opta stats
- Manchester United have come from behind to win nine Premier League matches against Southampton, an outright record in the competition.
- Since Ole Gunnar Solskjaer took charge, United have won at least five more Premier League points than any other side (32).
- Southampton are the first away team to score two goals from outside the box in a Premier League game at Old Trafford since Man City in September 2009.
- United striker Romelu Lukaku has been involved in 11 goals in his last 11 Premier League appearances against Southampton (9 goals, 2 assists).
- Against no side has Lukaku scored more Premier League goals than he has versus Southampton (9 - level with West Ham).
PSG, Arsenal and Wolves away in three different competitions represent a daunting two weeks to come for United. Southampton host Tottenham in the league next Saturday.