The Bears’ batting failed to really fire as they were dismissed for 221, with Michael Burgess (73) and Chris Benjamin (50) the main contributors.
But their bowlers fared better and with off-spinner Jacob Bethell claiming 4-36 on his white ball debut the remained in the hunt despite the efforts of Kiran Carlson (60) and Billy Root (50).
Lukas Carey was the hero at Sophia Gardens for the second time in three days as he once again hit the winning runs to snatch victory for his side.
On Tuesday it was his 29 not out that steered the Wales Minor Counties team to victory over Glamorgan in a friendly and two days later he hit successive boundaries to secure a two wicket win for the Welsh county over Warwickshire in the first round of the Royal London Cup with two balls to spare.
The Bears were defending a total of 26 runs in the final five overs and some brilliant death bowling by Ryan Sidebottom and Ethan Brookes looked as though it might pull off an unlikely win as they conjured up 11 dot balls.
A six from Roman Walker off Brookes in the penultimate over eased the pressure, but the game was still in the balance as Sidebottom roared in against Lukas Carey with five still needed from the last over.
Sidebottom started with two dot balls but Carey then reached the boundary with a far from convincing drive before walloping the next ball past the bowler to complete a two-wicket triumph.