The reigning champions of the trophy were 1/6 to win considering their great form and with Southampton potentially still reeling from their humiliating 9-0 defeat to Leicester last weekend.
Both sides will meet again at the end of the week at the Etihad again in a Premier League match-up.
Manchester City, as expected, were heavily dominating possession in the early stages of the game with the Saints setting up very defensively and looking to hit City on the break.
Ralph Hassenhutl’s side were starting to get into the game as they stifled the Citizens’ attack for the most part.
The likes of Riyad Mahrez and Angelino still had good potshots at Alex McCarthy but the keeper managed to stay strong and save them with relative ease.
Southampton were playing a back nine at one point as City were camped in their box with both their centre-back in the middle of the Saints half.
It was one of the centre-backs who opened the scoring for Guardiola’s team. In the 20th minute, Nicolas Otamendi nodded home a Bernardo Silva cross following a short corner in which the Saints had lost their concentration.
City were completely dominant throughout this half but could not create as many clear cut chances as Guardiola would have hoped for.
They did kick on from Otamendi’s goal, though, with Sergio Aguero curling a free-kick from a promising area just wide of the post.
The Argentine striker did not sweat the miss considering he scored just three minutes later from that chance.
It was a typical City goal, a slick team move with a cut-back from Kyle Walker for City’s record goalscorer to tap in for his 350th goal for the club.
It was no surprise that he would get on the scoresheet, as the bookies had him 2/5 to score anytime.
That was virtually the last big action of the game and as expected, City dominated with the Saints not even being able to get a shot off.
10 minutes after the restart, Aguero grabbed his second of the game. A Mahrez shot was saved and the 31-year-old reacted first to stab the rebound past the keeper.
Guardiola’s team looked to score even more goals, with Phil Foden’s cross unable to be converted by both Gabriel Jesus and Riyad Mahrez.
Later on, Southampton managed to muster up chances themselves. Winger Sofiane Boufal’s brilliant curling effort was saved by keeper Claudio Bravo, who virtually had nothing to do this game.
It was the Saints first effort on goal and it only took 65 minutes highlighting City’s dominance.
However, the travelling supporters did have something more to cheer about as Jack Stephens managed to grab a consolation. The defender headed in from a set-piece to give Southampton 15 minutes from time.
It was the last big action of the game as the Citizens closed out the game with ease and advance into the last eight of the Carabao Cup.
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