Summer signing Mohamed Salah was the hero for the Reds in front of the Anfield faithful, scoring twice in the space of 10 first-half minutes, before Philippe Coutinho made the result certain in the final half-hour.
The result saw the side move ahead of Arsenal into fifth in the table and, speaking to reporters afterwards, Klopp was full of praise for what he had seen.
"It was a difficult game, Southampton are a really good football side," he said. "It was an open game, we had more chances before we scored. We need to be patient in a lively way.
"The first goal, wow, I have nothing to do with the goal and the second was a fantastic pass and Mo Salah makes great runs. We could have scored more often but it was a perfect day for us."
Next up for the Reds is a trip to Sevilla in the Champions League on Tuesday night, ahead of a tough home encounter with Premier League champions Chelsea next Saturday.