The Merseyside club won their first league game since opening day over the Cherries, thanks to two goals from Oumar Niasse.
But Koeman believes there was no pressure for him to win the game in order to keep his job.
"For me, the players, for everyone it is important. Everyone knew it was a must-win game," said Koeman.
"Sometimes we go too fast - if you win you are sure about your job and if you lose you are not sure about your job, but that is ridiculous."