Weston turned sharply onto a Delroy Facey pass in the 39th minute before scoring with a low shot from 10 yards.
Notts County might have got another goal two minutes later when Jamie Forrester was brought down in the area by Bournemouth goalkeeper Shwan Jalal for a penalty. But Forrester's spot-kick was blocked by Jalal and Bournemouth escaped.
The Cherries should have been ahead in the first seven minutes but missed two clear chances. Notts County goalkeeper Kevin Pilkington made a double reflex save to deny Jeff Goulding and Anton Robinson and from the second rebound Brett Pitman clipped the crossbar.
Pitman was clear in the seventh minute from Mark Moseley's pass but shot straight at Pilkington from the edge of the area.
Pilkington had little else to do until the last minute when he plunged to his left at full stretch to keep out a fierce right-foot shot from 25 yards by substitute Shaun Cooper.