The Red Devils found themselves two goals down after 30 minutes of the Premier League clash at St Mary's through goals from Jan Bednarek and James Ward-Prowse.
United fought back in the second period, however, with star midfielder Bruno Fernandes pulling one back for the visitors just before the hour mark with his seventh league goal of the season.
Substitute striker Edinson Cavani levelled the scoreline 15 minutes later before sealing the three points with another goal deep in stoppage time.
The result was the eighth away win in a row in the Premier League for Ole Gunnar Solskjaer's team, setting a new record for the Old Trafford outfit.
It is also the fourth time in a row United have managed to claw their way back from behind to secure a victory.
Despite his side's struggles in the first half, Solskjaer said he always had faith that they could find a way back into the tie.
4 & 8 - Manchester United are the first side in Premier League history to win four consecutive away games despite trailing in each tie, celebrating eight straight away top-flight victories for the very first time. Backbone. #SOUMNU pic.twitter.com/uOFPmbz2Iy— OptaJoe (@OptaJoe) November 29, 2020
He told BBC Sport : "I was quite calm [at half-time]. It's about not conceding another goal and getting the next one.
"There was always going to be another one in that game. We must have had five or six clear-cut chances in the game today so it was about taking them when we got the chance. It worked.
"It is a trait of Manchester United to never give in. Every single one of them, there was absolutely no chance they were going to give up. That's a big thing for a Manchester United player and person."
United are now seventh in the English top flight and five points behind leaders Liverpool, though Solskjaer's men have a game in hand.
United sit top of Group H with nine points - three clear of their French counterparts - after four matches.