Junior Stanislas' first-half goal secured a first ever victory for the Cherries over the Blues and saw Koeman suffer his first defeat since replacing Roberto Martinez at Goodison Park in the summer.
Blues must start quicker
Stanislas' goal was the third game in five league games that Everton have fallen behind and Koeman said that his side could not continue to start matches in such sluggish fashion if they harboured ambitions of qualifying for continental competition.
Although Everton created the first big chance of the match, when Romelu Lukaku's header produced a stunning save from Artur Boruc at full stretch, the Blues found it "difficult" to play with the confidence they have shown in recent weeks.
Koeman admitted that Bournemouth were the "better team" and the home side produced six shots on target in the first 20 minutes alone, not including a deflected shot from Jack Wilshire, which cannoned back off the crossbar, before Stanislas curled home a spectacular effort from 25 yards to give the home team a 1-0 lead.
The Everton boss said after the match that his side's first-half performance "wasn't good enough", before adding that Bournemouth had "more aggression, won more second balls and put balls in the channels" than Everton did in the opening 45 minutes.
Improved second-half but Blues failed to find a way past the Cherries
Koeman was pleased with his side's "good reaction" in the second-half that saw the Blues become the "dominant and more dangerous team". The Dutchman said that "until the end I had the feeling that we could score" as Bournemouth "dropped back more and more".
In the second-half, Ross Barkley saw his powerful header go narrowly wide of the post, while substitute Enner Valencia almost bundled in an equaliser. Koeman was, however, left frustrated adding "it's a pity because you need to start like that as well" comparing his side's sluggish first-half to an improved second-half performance.