The Blades took a fortunate lead after just five minutes, Barrett putting through his own goal.
United went 2-0 ahead on 77 minutes and again it was due to an own goal, this time Cook the unlucky Cherries player.
Bournemouth 0 Sheffield United 2: Blades profit from own goals
Danny Wilson saw his Sheffield United return from Bournemouth in League One with a 2-0 win to continue their recent fine run.