The home side doubled their lead when Hanley rose highest in the penalty area from a Tom Cairney corner to power a header into the net.
It got even better for Gary Bowyer's men when Craig Conway surged forward and his effort took a deflection off Gestede which wrong footed Camp and allowed the ball to roll over the goalline.
Rhodes nearly notched his second of the game as he latched onto Marcus Olsson's cross from the left, but his shot smashed the inside of the post and bounced clear.
Callum Wilson then threatened the Rovers goal as he carved out space in the final third and curled an effort towards the bottom corner, but Robinson was on hand again to turn the ball wide.
Bournemouth did get a goal back in the 81st minute when Alex Baptiste brought down Adam Smith in the penalty area. Brett Pitman took the spot kick and fired the ball past Robinson to keep Eddie Howe's side's hopes alive.
Steve Cook got a second for the visitors in the 90th minute as he headed home from a corner, but Blackburn managed to hold out for the victory.