It was the Argentine's 150th league goal for City, a landmark that moved him level with Michael Owen in eighth place on the Premier League's all-time goalscoring list.
Saints did get one back on the half-hour mark, with Danny Ings converting from the penalty spot after the striker was felled in the box by Ederson.
Between them, the bottom seven of Crystal Palace, Burnley, Southampton, Newcastle United, Cardiff City, Fulham and Huddersfield Town have combined for just eight victories from 75 matches.
"I have the feeling he is now enjoying scoring goals and making assists".
With Liverpool only managing to draw against Arsenal on Saturday evening, Manchester City had the opportunity to establish a two-point lead at the top of the Premier League ahead of their clash with Southampton. The teams showed me some signals that in some positions we are not stable enough.
However, Sterling's first goal just before half-time set City up for a comfortable second half in which they scored two more - another from Sterling and a Leroy Sane strike.
Morata, the former Real Madrid and Juventus forward, has now scored five Premier League goals plus another in the Europa League this season.
Five minutes had gone when Aguero orchestrated an attack on the edge of the Southampton area, David Siva then moving the ball on to the overlapping Sane, whose cross was turned into his own goal by Saints' Wesley Hoedt.
"We played excellently without, and especially with, the ball", said Guardiola. The game was not over.
Unbeaten Chelsea made a slow start but Morata cheered the Stamford Bridge crowd by putting his team in front from close range in the 32nd minute after pouncing on a cross by Pedro. 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.
And a 3-0 win over Shakhtar in Ukraine two weeks ago means City top their Champions Leaue group, and are one point clear of Lyon.