Robin van Persie was the beneficiary of some generous Southampton defending on Monday night, scoring a goal in each half to extend United's winning run to five matches.
But Louis van Gaal's team found themselves on the back foot for long periods at St Mary's.
Graziano Pelle scored his 10th goal of the season to equalise in the 31st minute but they slumped to their third loss in nine days, after Van Persie reinstated United's lead in the 71st minute.
"We did everything, we created chances. But we also did mistakes," Koeman told a news conference.
"Maybe they [United] did more mistakes than we did but they have players who handle those kind of mistakes and that was the difference today.
"I said to my players, we lost not against Man United tonight; we lost by ourselves."
Koeman felt Van Persie's second goal had a debilitating effect on his team as they struggled to make a convincing case for a second equaliser.
"It is always difficult because you are the better team and you have to fight again against 2-1 down," he said.
"After that moment we didn't play well but thats the difference. At this level details make the difference in football."
Despite their recent slump, Southampton remain fifth in the Premier League and Koeman is committed to the style of play that has brought success to the south coast club over recent months.
"I like to see my team playing football in the way that we've started this season,” he added.
"If we lose or we win, that's our way of playing because I believe in that way of playing even now when we've lost the last three games.
"City were the much better team but we deserved more against Arsenal, we deserve much more tonight but we did mistakes.
"If you give players like Van Persie those kind of mistakes, he will punish you - he will punish you for sure."
