Bournemouth were ahead in the 16th minute when Crawley striker Gary Alexander glanced home a centre from Steve Cook for a spectacular own goal.
Striker Brett Pitman fired Bournemouth's second in the 40th minute when a quickly-taken 25-yard free-kick caught Crawley by surprise.
Crawley substitute Billy Clarke was denied by the legs of Bournemouth goalkeeper David James before the home side made sure of the points with a third goal in the 84th minute.
Simon Francis spilt the Crawley defence with a deep cross from the right and Eunan O'Kane ran clear to beat goalkeeper Paul Jones.
Crawley goalkeeper Paul Jones prevented Bournemouth winning by a bigger margin with saves from leading goalscorer Lewis Grabban and from a Cook header in the second half.