In yesterday's other game, second-place Chelsea breezed past Crystal Palace 3-1 at Stamford Bridge.
City needed only five minutes to take the lead against Southampton, Wesley Hoedt putting through his own net. Sterling then quickly laid on the second as Aguero became the ninth player to reach the landmark of 150 Premier League goals.
It was proving a nightmare return for Southampton manager Mark Hughes, who was in charge at City during 2008-09, as his side conceded a third when Sterling supplied a cross and David Silva lashed the ball home.
Southampton, without a win in six league games, replied in the 29th minute when Danny Ings won and converted a penalty past Ederson.
But City restored their three-goal cushion on the stroke of halftime with Aguero taking advantage of a mistake by Cedric Soares and crossing for Sterling's fifth league goal of the season. Sterling added his second of the game midway through the second half before Sane completed the rout in stoppage time.
"We played excellently without, and especially with, the ball," said Guardiola. "We were aggressive in the final third, but we were lucky to score the goal in the last minute of the first half. The game was not over. Southampton created more chances than most teams this season."
Alvaro Morata continued his recent resurgence with two goals and fellow Spaniard Pedro added a third as Chelsea proved too strong for Roy Hodgson's Palace. Unbeaten Chelsea made a slow start but Morata cheered the Stamford Bridge crowd by putting his side in front from close range in the 32nd minute after pouncing on a cross by Pedro.
Palace, beaten in their previous five league matches, levelled early in the second half when James McArthur sent Andros Townsend clear, and he slammed a right-foot shot into the net.
Chelsea reacted by bringing on dangerman Eden Hazard and Mateo Kovacic, and within minutes, were back in front. Hazard took a free kick on the right and the ball eluded several players before reaching the unmarked Morata at the back post, and he struck his fourth goal in his last four league games.
Pedro added the third as Chelsea moved on to the same points tally as Liverpool, two behind City.
- AP