The south-coast club extended their unbeaten run in the league to four matches thanks to goals from Sadio Mane and Dusan Tadic – albeit aided by some calamitous Arsenal defending.
Mane produced a stunning finish from a tight angle after Laurent Koscielny had slipped, and Tadic doubled their lead after Mathieu Debuchy and Wojciech Szczesny failed to deal with a low cross.
Koeman was delighted with the hosts' ruthlessness and revealed that Sunday's 1-1 draw against title hopefuls Chelsea had given his charges renewed belief.
"I think it was a fantastic performance from the team," he said. "We had very good organisation and had the best chances in the game. We had to kill it after 2-0 and in several moments our defending was outstanding.
"We had to score more. It's strange to beat Arsenal 2-0 and you're a little bit critical as we should have had three or four more at least.
"We have a good team and can beat the big ones if we show the spirit we did today, everything is possible.
"After the draw against Chelsea the players had the belief to win something, today we made less mistakes and that's the big difference."
The win moves Southampton three points clear of Arsenal, strengthening their grip on fourth place and the final UEFA Champions League spot.
