The visitors were first to bat and were kept at a reasonable rate thanks to some good bowling from the Warwicks attack.
Neil Carter was the first Warwickshire bowler to take a wicket when he had Durston caught by Chris Woakes. Carter followed up this up by getting Bosman caught by Jim Troughton.
Troughton caught Hughes to leave Derbyshire on 22-3 but Park and Smith came together to put on 63 for the fourth wicket.
John Sadler finished 38 not out as the visitors ended their twenty overs on 148-7.
Neil Carter and Ian Bell both were out for two runs before Jon Trott (44) and Jim Troughton made a third wicket stand of 67.
When Troughton was out for 58 from 41 balls including two sixes, 28 runs were still required by a Warwickshire team which was limping along.
It went down to the last ball but Darren Maddy (25 not out) and Rikki Clarke (7 not out), steered Warwickshire to a six wicket victory.
Warwicks continue in Twenty/20 action on Sunday with a match at Trent Bridge against Nottinghamshire. Play is due to start at 2.30pm.