The Blaze bowled with discipline to restrict the Bears to 158-8.
Openers Abi Freeborn (56) and Meg Austin (44) added 95 but wickets fell freely thereafter against an attack led by former Bears seamer Grace Ballinger (3- 24) and Lucy Higham (2-16).
The Blaze then reached 159-6 with four balls to spare thanks to Marie Kelly, another ex-Bear, who scored 45 at the start and a robust unbeaten 27 from Emma Jones at the end. Amu Surenkumar took 2-19.
The Blaze have started the Blast superbly but Warwickshire, finalists last season, are bottom of the table after three straight defeats sustained by a squad stretched to the limit by injuries and international calls.