After being so resolute defensively last campaign, Southampton have started this season conceding five goals and attaining just one point from their opening two games.
They did not look any more threatening in attack against promoted Watford at Vicarage Road, but a first clean sheet of the season has given Koeman reason for optimism at Southampton.
“We played a little bit different than the first two games because we had to be defensively better,” the Dutchman told BBC Sport. “In my opinion, it always starts at the back and at least we had our first clean sheet of the season.
“One of the key points of last season was that we had a strong defensive organisation.”
Koeman was then queried on the status of Sadio Mane and Victor Wanyama, with both players having been linked to Manchester United and Tottenham respectively with just one week of the transfer window remaining.
“We know have to deal with speculation over our players,” Koeman said.
“Again, no-one is for sale. Everyone stays at Southampton. We want to bring in another defender,” he added, with rumours surrounding a potential £10million move for Celtic centre-half Virgil van Dijk.
Midfielder Wanyama was reported to have requested a transfer over the weekend, rumours that Koeman was keen to play down.
“All I can say is that for Victor, the future is in Southampton,” he told Sky Sports. “And for every player that we like to stay, they have to stay.
“And it is all about contracts and you have to respect contracts and nobody is for sale. And I spoke about that I think two or three days ago, and it still has the same meaning.”
Koeman moved to alleviate fears over Mane, who was substituted after suffering a concussion following consecutive clashes with Heurelho Gomes and Troy Deeney. The Southampton boss expects the Senegal international to be available for the Europa League qualifier second leg against FC Midtyjylland on Thursday.