Everton manager David Moyes felt his side were "unfortunate" to come away from Burnley without at least a point on Sunday.
The Toffees lost 1-0 at Turf Moor after conceding a 34th-minute Wade Elliott goal.
Martin Paterson also hit the crossbar with a header in the opening minute, while Louis Saha missed a second-half penalty wrongly awarded when Tony Hibbert went down in the area.
It was Burnley's second successive 1-0 home win after defeating champions Manchester United on Wednesday.
"Burnley were the better side in the opening 25 minutes. I think, with the confidence they gained from the game in midweek, we expected that from them," added Moyes.
"But after that I thought we got back into it and were a little bit unfortunate not to get something out of it."