Van Dijk, a €17million deadline-day signing from Celtic, headed Saints’ opening goal and kept Swans striker Bafetimbi Gomis so quiet he was hauled off at half-time, and then did a similar job on substitute Eder in the second half.
Koeman said: “One of his qualities is he is a good header of the ball. He is also defensively strong and his personality is important for the team.”
“I’m very pleased,” Van Dijk told Saints’ official website. “If you look at my height I should be dangerous every time. We practice it a lot.”
Dusan Tadic added the second goal with the aid of a deflection just after half-time and Sadio Mane slotted in the third on the hour.
Gylfi Sigurdsson grabbed a late consolation from a penalty but Swansea were never at the races to the annoyance of manager Garry Monk.
“We were not at the right levels,” he said. “We are making errors which is uncharacteristic because we’ve shown already this season the standards we can reach. We have to be consistent. The last two away games have just not been good enough.”
Stekelenburg, Soares, Fonte, van Dijk, Bertrand, Ward-Prowse, Wanyama, Mané (Juanmi 86), Davis (Romeu 63), Tadic (Rodriguez 74), Pellè Subs not used: Davis, Yoshida, Long, Rodriguez, Romeu, Martina, Juanmi.
Fabianski, Naughton, Fernandez, Williams, Taylor, Ki Sung-yueng, Cork (Montero 45), Shelvey (Britton 80), Ayew, Sigurdsson, Gomis (Éder 45).
Britton, Nordfeldt, Routledge, Éder, Montero, Rangel, Bartley.