The Blues went down to a 91st minute winner from Scott Arfield, who was lucky to be on the pitch after several fouls which could have resulted in a second yellow card.
Everton dominated the rest of the game, however, and should have been in the clear had Lukaku been able to get on the end of crosses from Bolasie and Deulofeu.
But Koeman was left rueing another away defeat to a side we should be beating.
“It is hard to take because sometimes football can be so unfair,” said the boss.
“We did not deserve to lose. We did so many good things but finally the result is not what we want.
“We started well, maybe we dropped a bit for the last 20 minutes of the first half. We played well in the second half, we had a good reaction and had enough chances to score.
“Burnley were very well organised and dropped back, it was difficult but I thought we played well in that time.
“To travel back without any points is really very disappointing. It’s one of those days when you cannot believe you lost the game.”