City's shock run of three defeats in four league games has allowed the unbeaten Liverpool to close in on a first title since 1990 with the sides set to meet at the Etihad on Thursday.
But after Jurgen Klopp's men thrashed Arsenal 5-1 on Saturday to open up a nine-point lead, the champions responded despite an early scare at St Mary's.
"We knew it, that with the position of Liverpool, if we drop points then it is over, it is finished," said City boss Pep Guardiola.
"It would be almost impossible. Of course they are going to drop points but not too many."
Charlie Austin wasted a great chance to give Southampton the lead before David Silva slotted home Bernardo Silva's cross to put the visitors in front.
However, just as in defeats to Crystal Palace and Leicester, City surrendered the lead when Pierre-Emile Hojbjerg blasted high past Ederson eight minutes before half-time.
The game swung back in back in City's favour just before the break as James Ward-Prowse turned Raheem Sterling's cross into his own net before Alex McCarthy couldn't keep out Sergio Aguero's header.
Guardiola's men should have added to their advantage as Aguero hit the bar and McCarthy made amends by denying Gabriel Jesus.
Fernandinho had been badly missed in City's recent defeats, but his return from injury made a huge difference in midfield and Southampton ended with 10 men after Hojbjerg was shown a straight red card for scything down the Brazilian five minutes from time.
- Pogba doubles up -
Pogba had scored just one goal from open play in the Premier League this season under Jose Mourinho, but continues to flourish since the Portuguese's departure with his second double in four days.
"It is different, we still won games with the old manager but it is just a different style of playing, we are more offensive and we are creating more chances and that is how we want to play," said Pogba.
The Frenchman's opener on five minutes owed much to the brilliant skill of Marcus Rashford who dribbled beyond two Bournemouth defenders before crossing for Pogba to tap into an empty net.
Pogba then powered home a header from Ander Herrera's inviting cross before Rashford's outside of the boot finish put United 3-0 up.
Nathan Ake reduced Bournemouth's deficit before half-time, but Pogba turned provider for substitute Romelu Lukaku to make it 4-1 and only the post denied Pogba a first ever hat-trick.
"That is a top, top performance for a midfield player because it's all-round," said Solskjaer of Pogba's display.
Despite Solskjaer's 100 percent record, United remain eight points off the top four as Chelsea ground out a 1-0 win at Crystal Palace thanks to N'Golo Kante's winner.
The French World Cup winner has been encouraged by Maurizio Sarri to get into more goalscoring positions with Jorginho occupying Kante's usual holding midfield role for the Londoners this season.
"He is improving, especially in his movement without the ball," said Sarri.
"Today his movement was really very good and done with the right timing."
Southampton are now above the relegation zone only on goal difference as third bottom Burnley secured just a second win in 13 games by beating West Ham 2-0.
Sean Dyche dropped former England number one goalkeeper Joe Hart to the bench for the first time in the league this season and was rewarded with a rare clean sheet as first half goals from Chris Wood and Dwight McNeil secured a vital three points.