2-1 down at St Mary's, Mohamed Salah struck twice to earn the Reds the points.
Slot said afterwards: "In the second half I saw the intensity I was looking for in the first half. Southampton make it really difficult for many teams, including top clubs, and they made it difficult today, though we dominated the game completely in my opinion.
"There was not enough urgency to arrive in the box (in the first half), we just kept the ball instead of attacking the box. Although we fell behind in the second half we showed more urgency and intensity. That led to numerous chances and eventually two goals.
"It is tough to create chances against a 4-5-1 low block but we know Southampton want to play out from the back so you have to be really aggressive, if not they have a really good game plan".
On Salah, Slot continued: "If you want to win you need to score goals and we know Mo can score. For me his first goal is the most important as we scored out of nowhere, then after that it was just waiting for us to score another.
"The positive is we fought back, but it was only two times they arrived in our 18-yard box. In general we dominated but because of the scoreline it felt like a difficult one, but we deserved to win.
On facing Real Madrid and Man City this week, he added: "They are two teams who have dominated football the last few years. So two big games and that's why we needed to win today and that is what I am happy about."