The Cherries went in front with just three minutes gone when Brett Pitman's inswinging free-kick avoided everyone, including keeper Jamie Spencer, and ended up in the back of the net.
Despite Chester pressure, they could not find a way past a stubborn Bournemouth defence.
Ryan Lowe lofted the ball onto the top of the net from 10 yards out, and Kevin Roberts was within inches of getting on the end of a corner when a connection of any kind would have brought a likely goal.
Mark Molesley had a couple of good chances to make the game safe, on one occasion seeing a superb effort saved by Spencer after he had created the chance himself with a superb bit of individual skill.
But it was left to Anton Robinson to secure the win, poking home from close range in the 79th minutes after Spencer had saved sub Liam Feeney's initial effort. The defeat leaves Chester four points from safety with just two games to play.