Bournemouth manager Eddie Howe said: "It was not a great game, the conditions played a part in that.
"Burnley did what they do, we expected the game that it was but we need to do better. It was a much better second half and we were the better team but could not break down a physical defence.
"It is how they play, it is up to the referee to control the game. We paid the price for not marking from a ball into the box.
"We feel hugely disappointed to lose the game. The first half was very scrappy, to lose the game is a devastating blow. It was not one for the purists, a lot of free-kicks given away and you know every free-kick will come into your box.
"There were a lot of challenges in the game that need reviewing."
Burnley manager Sean Dyche said: "It was super poor conditions, a poor game but we stuck to the task. It is hard to find ways of winning and we managed to do that.
"The pleasing thing is, in the second half they were better than us, but we stayed resolute to the task.
"[Nick] Pope has had minimal to do regarding saves and we found a moment. There was not a lot of goalmouth action, not a great game but we get three points. We hope and want to play better but we get the three points.
"What a header from J-Rod. We know we have to play better away. At this stage of last season we were on 12 points, it was really ugly, but we managed to find a way."