Jeff Hendrick's first-half goal ensured the Merseyside club have only won once since the opening day of the season.
Koeman's Toffees are now just two points above the bottom three and pressure is growing on the Dutchman.
He said: "I'm not unhappy what I saw from the team, they showed today how we had to play.
"If you play against a team who are defending so strong you need to be at your best. We didn't have that creativity."
Wayne Rooney thought he should have had a penalty in the closing stages when his flick-on from Gylfi Sigurdsson's cross bounced on to Matthew Lowton's arm.
Koeman said: "You know, sometimes you get the help of a referee.
"I don't say it was a penalty because it wasn't his intention to touch the ball - but on other occasions I've seen a penalty given.
"Of course we can play better on the ball, this situation is really disappointing. But we'll continue because what I saw today is how to come out of this difficult situation."
While Everton’s struggles continue, Burnley have now gone five Premier League games unbeaten.
Clarets boss Sean Dyche said: "I'm very pleased. I know there's a bit of noise round here about how they're playing but there are still some very good players [in the Everton squad].
"We had a soft start for 10-15 minutes, then scored a fantastic goal.
"I've spoken about the flexibility of the side. We can play when we need to, and be longer and stronger when we have to - but we can defend too. It's about finding that mixture."
