Until Stone’s final over, Yorkshire were heading top of the group after a superb run-chase at Headingley.
Alex Lees blazed 73 from only 35 balls with six sixes to lead Durham to 205-9, with Ben McKinney also making 42 from 21 deliveries.
Captain Jonny Bairstow got the chase off to the perfect start with 66 from 36 at the top of the order before Moeen Ali kept them alive by swinging seven sixes in his 60 from 32 balls.
When Moeen fell, 38 were needed from 23 balls and it needed Matthew Revis to lift Kasey Aldridge down the ground for six to win the game off the penultimate ball.
In the South Group, Essex moved third – trailing second-placed Surrey only on net run rate – with a totally dominant performance at Hove.
Michael Pepper made 47 from 29 balls after winning the toss before Simon Harmer dismantled Henry Crocombe’s final over of the innings, taking it for 31 runs as Essex posted 204-6.
It knocked the wind out of Sussex as their chase never materialised and Sam Cook picked up 4-7.
A seventh defeat of the season keeps Sussex fifth in the table.
There was also a blowout victory for Worcestershire at New Road in a cross-pool fixture as they met Kent for the very first time in a T20.
Kent won the toss and held the home side to 142-7 with captain Adam Hose top-scoring with 47 from 41 balls.
But just two players managed double-figures in the reply as Kent were rolled in only 14.1 overs.
Kent, in the South Group, missed the chance to join Essex and Surrey on 20 points and remain fourth, while Worcestershire, in the Central and West Group, did improve to 20 points and are now outside the top two only on net run rate, which was boosted by such a large win.
Glamorgan are also on 20 points in the Central and West Group after chasing 185 to beat Warwickshire at Cardiff.
Sent in, Sam Hain scored 63 from 46 balls for the visitors and Australian Beau Webster 47 from 32 as the Bears reached 184-4.
The chase was always comfortable after Will Smale, who made 62 from 50 deliveries, and Kiran Carlson, put on 75 in the powerplay.
Asa Tribe (38 off 31 balls) and Sean Dickson (24 off 11) then got the hosts home with seven balls to spare.
A sixth defeat for Warwickshire leaves them fifth in the group on 12 points with three games remaining.