Chasing a target of 174, the Foxes self-destructed, losing eight wickets for 52 runs to collapse from 97 for two to 149 all out in 18.2 overs.
It was their second defeat in 24 hours and they now face an uphill battle to qualify for the knock-out stages.
Josh Cobb hit a quick-fire 47 off 26 balls with six fours and a six to give the Foxes a flying start to the chase as he dominated an opening stand of 54 in six overs with Niall O'Brien.
Even when O'Brien was bowled trying to sweep Dan Redfern, the Foxes still looked on course at 87 for two. But a breathtaking catch at cover by David Wainwright to dismiss Bangladesh star Shakib Al Hasan off the bowling of Albie Morkel proved the turning point.
Although Australian Joe Burns made 36 off 25 balls, hitting three fours and two sixes, he found little support from the rest of the side with wickets falling in clusters.
In addition to his brilliant airborne catch, Wainwright claimed three wickets for 24 runs in his four overs and Redfern and Morkel picked up two each.
Put into bat, Derbyshire posted 173 for three with Durston leading the way. He shared an opening stand of 48 with Chesney Hughes and then a second wicket partnership of 94 with Shiv Chanderpaul.
Durston paced his innings superbly, hitting five fours and three sixes before finally being dismissed in the 20th over with a leg-stump yorker from Robbie Williams.
Hughes (34) and Chanderpaul (38) kept the Falcons innings ticking over but the Foxes restricted them to 40 runs off the last five overs.