Bournemouth made the brighter start with Liam Feeney providing a threat down the right flank and his delivery found Brett Pitman who headed over the crossbar.
Accrington struggled to get going and their frustrations were compounded in the eighth minute when manager John Coleman was sent to the stands after protesting to referee Michael Oliver after Michael Symes was booked for a foul on Warren Cummings.
But the home side ended the half strongly with Bournemouth goalkeeper Shwan Jalal saving Jimmy Ryan's free-kick and a header from Robert Grant.
John Miles sent Symes through on goal early in the second half, but the striker failed to beat Jalal from 12 yards, while referee Oliver waved away appeals for a penalty when Jason Pearce challenged Symes in the area soon after.
Bournemouth grabbed the winner in the 76th minute when Cummings' free-kick was swung towards the far post and Hollands stooped low to glance a header past Dunbavin.