The Clarets took the lead midway through the second-half courtesy of Ben Mee's headed goal, but went on to concede three within the space of six minutes as the Toffees stormed their way back and claimed all three points.
Speaking after the game, Burnley boss Sean Dyche said: “I was very pleased up until their goal with the performance, and the way we controlled a lot of the game away from home, which is difficult at places like this where they have had a good start.
“We concede an awkward first, we are a bit misshapen and haven’t come out of the set piece situation. After that it is the madness of football and we have a mad six minutes and the game is gone.
“That can happen in the Premier League but it is something we have to get a grip of because it has happened twice with the Brighton game, but within that there is a little bit of wanting that first win.”
He continued: “It is not quite there, you go 1-0 up and that feeling where you go on the back foot rather than the front foot and the message to the whole group is make it happen and not wait for it to happen.
“The players here know that but they have to make it happen.”
Burnley will look to bounce back this weekend, when they host Arsenal at Turf Moor on Saturday afternoon.
Take a look at the latest news round-up from the Premier League, as the build-up to the weekend's action continues: