Southampton beat Chelsea 3-1 at Stamford Bridge in their most recent fixture.
The Saints went into the break level at 1-1, after scoring through Steven Davis, but Koeman did not want his team to simply settle for a point.
Instead, he urged them to go for the jugular, and he got the response he wanted, with Southampton going on to score twice more in the second-half to wrap up a comprehensive 3-1 victory.
“We spoke at half time and we said, ok, important now is we try to attack and try to win the game and no defending the 1-1,” Koeman explained.
“I think we showed from the beginning in the second half we pressed, the team was very compact and it was difficult for them.
“Good football, great movements and good goals, and maybe we had to score one or two more in that first 30 minutes in the second half.”
Koeman's half-time team-talk showed the ambition that Southampton have, and it proved just how vulnerable Chelsea are at the moment.
Stamford Bridge used to be an impenetrable fortress for away sides, but Chelsea's poor form has meant that any visitors there feel that they have a chance of victory.
Southampton's win wasn't achieved through backs to the wall defending, either, as Koeman's side dominated the match for large periods.
The Saints attacked with patience and poise, with the performances of Graziano Pelle and Sadio Mane being particularly eye-catching.