Hojbjerg, who has had a minor history of reckless mistakes, was sent off five minutes from full-time for a lunging challenge against Fernandinho.
The Dane had superbly cancelled out David Silva’s opening goal, before a James Ward-Prowse own goal and Sergio Aguero tallied to lift the defending champions back into second place.
City dominated throughout owning 77 percent of the possession and nearly took a lead within five minutes, only Silva to be denied from point-blank by Alex McCarthy.
But the midfielder got the better of the Saints’ goalkeeper moments later when he was left open in the box to hammer a shot past McCarthy.
Riyal Mahrez failed to double the lead when he curled an effort wide of the post and Aguero fashioned a chance that required a save.
Against the run of play, however, Southampton almost equalised when Charlie Austin met Ward-Prowse’s corner-kick, but Ederson made a reactionary save to prevent the goal.
But Southampton took advantage of an error when Oleksandr Zinchenko was caught in possession and Hojbjerg fired beyond a flat-footed Ederson.
Southampton had a strong penalty appeal waved away when Ward-Prowse was bundled over in the box by Zinchenko.
The decision proved to be crucial for Southampton who went behind 2-1 as Raheem Sterling’s cross took a wicked deflection off of Ward-Prowse.
City then made it 3-1, with Aguero heading in from Zinchenko’s cross and the damage was done.
Southampton: McCarthy, Targett, Bednarek, Stephens, Ramsay, Ward-Prowse, Hojbjerg, Lemina (Redmond 46), Romeu (Valery 59), Elyounoussi, Austin (Long 68).
Unused subs: Gunn (GK), Yoshida, Armstrong, Ings.
Goals: Højbjerg (37)
Bookings: Austin (58), Bednarek (73), Højbjerg (RED 85).
Manchester City: Ederson, Danilo, Kompany, Laporte, Zinchenko, Fernandinho (Walker 86), Silva, Bernardo, Mahrez (Sane 84), Sterling, Aguero (Jesus 73).
Unused subs: Muric (GK), Stones, Otamendi, JFoden.Goals: Silva (10), Ward-Prowse (OG 45), Aguero (45+3).
Bookings: Aguero (32), Kompany (72), Danilo (78).