At Chester-le-Street, opener Bates hit 12 fours for the hosts and shared a stand of 73 with Mady Villiers, who made 36 from 24 balls.
The pair led their side to 80-2 in the 10th over before wicketkeeper Bess Heath hit six boundaries in amassing 45 runs from 27 balls to lift Durham to a decent total.
Despite losing Marie Kelly in the first over of the chase, The Blaze were always up to their task, largely thanks to a Beaumont innings featuring 11 fours and three sixes.
They needed 14 runs from the final 12 balls but Graham took consecutive fours off Lauren Filer before Georgia Elwiss (24no) struck the winning boundary.
The win was The Blaze’s fourth of the season and keeps them top of the table.
