Koeman's men created a number of chances at St Mary's on Sunday to level the scoring after Charlie Austin's first-minute opener, but Everton's finishing let them down as they failed to find the net.
"Obviously after they scored within the first minute, they will then drop back, defend and wait for the counter and that is what they did," Koeman told BBC Sport.
"Having said that, I cannot complain about our second half, but if you do not score, you cannot win.
"We had good chances but they had better chances to score and our start was really poor.
"We know Saints are a good team, they play in Europe and have very good players.
"Their system has not changed much, the goalkeeper and defensive players are the same and are still strong."
Southampton boss Claude Puel, meanwhile, saw plenty of reasons for optimism with his side's performance
"It was a very good performance for us because we had a lot of injury problems before the match," the Frenchman stated.
"But it was a good game against a good team, it had lovely action and I took pleasure from our chances, it is interesting for the future.
"It is always better to start with a goal as normally it is us conceding early. Our performance was good for our confidence after the loss against Sparta Prague earlier this week."