Meanwhile, Saints are now seven Premier League games without a victory and lie 16th.
However, it also marked Aguero's 150th goal in the Premier League.
City may have taken the lead after just six minutes, as Hoedt's attempted clearance ended up in his own net, but Southampton had already been warned of their hosts' threat on several occasions.
"That's the only chance Liverpool have got, because City are better than them".
City doubled their advantage when Sterling carved his way into the area and squared for Aguero to fire home another landmark goal.
Belgium global Hazard returned from two weeks out with a back problem when he came on in the 64th minute of Sunday's 3-1 victory over Crystal Palace.More news: New U.S. Sanctions Against Iran Go Into Effect
Manchester City, which has yet to lose a Premier League match, tops the standings with 29 points, just two points ahead of its closest rival, Liverpool, which was held to a 1-1 draw against Arsenal on Saturday night.
Palace started the game in brighter fashion.
Asked if it would be better to play Morata to help maintain his resurgent form, the Italian added: "No".
The Citizens have won eight of their opening 10 matches, drawing the other two, and are out in front by virtue of their superior goal difference compared to Liverpool's. Willian then saw a goal disallowed for offside in the closing stages of the first half. For us, the best work for him is that he will play for 45 or 50 minutes tomorrow [on Thursday].
Morata would have been happier to have taken the matchball home and he was visibly annoyed after Palace goalkeeper Wayne Hennessey read his attempted lob when through on goal in stoppage time at the weekend.