Bury created plenty of chances in the first half, but found themselves two behind at the interval.
Summer capture Ryan Lowe saw a goal chalked off for offside, before Bournemouth went in front courtesy of a cracking goal out of nothing as striker Brett Pitman turned and lashed a shot home from 20 yards via a post.
Brian Barry-Murphy curled a free-kick wide for the hosts, but they were undone by a simple goal five minutes before the break as Pitman crossed from the right and Steve Fletcher nodded home to make it 2-0.
The Cherries went further in front five minutes after the restart with another superbly-struck effort as Mark Molesley powered home an angled shot from 25 yards.
Lowe fluffed a great chance for a consolation late on when he side-footed straight at goalkeeper Shwan Jalal.