After fellow Argentina international Nicolas Otamendi headed City in front in the 20th minute, Aguero struck in either half to move on to 12 goals for the season.
Despite Jack Stephens' late consolation goal, it was a pretty routine win against a Southampton side still reeling from Friday's 9-0 home defeat to Leicester, but one City fans will remember as a famous name from their past emerged as the next youngster hoping to be part of the future.
Boss Pep Guardiola made nine changes to the City side that beat Aston Villa on Saturday, bringing in teenagers Phil Foden and Eric Garcia, but the most notable was a debut for 18-year-old Tommy Doyle.
The midfielder certainly has the pedigree with both his grandfathers – Mike Doyle and Glyn Pardoe – part of the City side that won the old First Division title in 1968, the FA Cup in 1969 and the League Cup in 1970.
Mike Doyle lifted the League Cup again in 1976 as captain, the last major trophy City won before the Sheikh Mansour era.
Of course, times are different now and it is three years since City last lost a League Cup tie, winning the trophy in each of the past two seasons.
That record never looked like changing as City enjoyed 77 per cent possession and Southampton did not even manage a shot on target until the 65th minute.
By then, City were in complete control. The opening goal came in the 20th minute, with Doyle's short corner played back for Bernardo Silva to loop in a cross, and Otamendi beat team-mate Gabriel Jesus to head it in.
After relentless City pressure, the second came in the 38th minute. The hosts shuffled the ball over to the right channel and into the path of Kyle Walker, whose low cross left Aguero with the simple task of side-footing home from close range after he dropped off his marker.
By the 56th minute, it appeared over as a contest. Riyad Mahrez's shot was deflected by Jan Bednarek, but looped up and Aguero got his toe to the ball before Alex McCarthy to slip it into the net.
Mahrez then somehow missed a fourth, denied by McCarthy at point-blank range from Jesus' lay-off.
It should have been game over regardless, but Southampton were determined to show some kind of response to Friday's horror show.
Their first shot finally arrived in the 65th minute, with Sofiane Boufal surging forward and hitting a low shot which Claudio Bravo pushed around the post before Shane Long headed over from the resulting corner.
And they had a goal in the 74th minute when Stephens rose to head home James Ward-Prowse's corner.
That, and a spot of cramp for Angelino, prompted Guardiola to abandon plans to bring on youngsters Adrian Bernabe and Ian Poveda-Ocampo as John Stones was introduced instead and City found themselves defending late on.
Garcia, the 18-year-old Spaniard, moved over to right-back in the reshuffle and continued an impressive display in which he looked strong and composed when called upon.
Given City's injury worries at the back, he could well be seen in the Premier League before long.