The south coast club were only trailing by a single goal when the visitors had centre-back Eliaquim Mangala sent off with still 15 minutes remaining at St Mary???s.
However, instead of taking advantage of the champions??? numerical disadvantage, the Saints conceded two more goals to see their opponents move above them into second place in the table.
And after the match Koeman could not hide his frustration at seeing Southampton???s unbeaten home run in the league this season come to an end.
???I am disappointed as I think in the second half at the beginning, it was a little bit better than at the start of the game,??? said the Dutchman.
???And you know that if you make any mistakes you will be punished, but it was not good enough today and we have to realise that. In the beginning we were too nervous, City were the dominant team, we lost some stupid balls in midfield, but I think that after 20 minutes we controlled it a little better, we had a good organisation.
???We had a good chance in the first half, they had a good chance as well, and then in the second half mistakes,??? added Koeman, ???and you know that you cannot even make mistakes against nine players ??? they scored twice against 10 men ??? and that is too much and at this level you will pay for that.???
However, in the end Koeman did not have any complaints about the final result, with the Saints boss recognising City???s greater class on the day.
???The difference between the two sides today was their quality, and we knew that we would have to be good in all the aspects of our game, and it was not like that,??? he said.
???We know that in three days we have to play Arsenal away, so there is no time to be disappointed. You have to learn about the mistakes of today, you know that if you want to beat these big teams, you have to play better.
???And that is the message that I said to the players.???