Despite dominating proceedings on the south coast, second-placed Southampton had to wait until the 75th minute to take the lead through substitute Shane Long's first league goal of the campaign.
Long wrapped up the points five minutes later to secure Southampton's fifth consecutive win in all competitions, and Koeman was delighted with how his team calmly waited to break down the visiting defence.
"We had enough patience today," he told reporters. "It was difficult, we missed some quality in the final pass in the first half.
"We were the dominant team in the game but we didn't create a lot of opportunities and we spoke about that [at half-time].
"It's another win and another clean sheet and sometimes it's difficult because maybe people expect an easy win with the position in the table.
"If we keep the good organisation we will score, because we have the players who make the difference."
Long joined Southampton in an estimated £12 million deal ahead of the season, but has struggled to nail down a regular spot, starting half of his 10 league appearances.
And Koeman was delighted to see the Republic of Ireland international leave his mark on the contest.
"Shane is doing very well," the former Feyenoord coach added. "They [Leicester] had good organisation but today he made the difference when he came in.
"In the beginning he had some adaptation problems with how we play and how we train.
"It's fantastic for me if you change a player and he scores two times but that's not always the case, but that shows it's really a good team. It's not about 11 players it's about the whole squad."
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