Heavy rain seemed to have scuppered any hope of play on a night just three matches from seven got under way, and only two ended with a positive result.
The Jets chased down Leicestershire's 88-2 to win thanks to Paul Coughlin (38 not out) and Paul Collingwood (34).
In the South Group, Hampshire beat Gloucestershire by 24 runs in Bristol.
Gloucestershire scored 175 in 19 overs, with Australian Michael Klinger hitting a magnificent unbeaten 101 from 58 balls, which included six sixes and seven fours.
But openers James Vince (42) and Rilee Rossouw (45) ensured the visitors were always ahead of the run chase.
The reached 121-3 to earn a 24-run victory courtesy of the Duckworth-Lewis-Stern system when the rain forced an early end.
The victory lifted Hampshire four places to third in South Group.
Durham Jets appeared to be facing a seventh straight defeat in this season's competition when a Luke Ronchi-inspired Leicestershire made 88-2 from their seven overs at Grace Road.
The New Zealander smashed a six and five fours from James Weighell's first over and went on to make 63 from 21 balls.
Despite losing both openers cheaply, former England batsman Collingwood and Coughlin teamed up to guide the away side to an unlikely victory
However, there was drama in the final over as Collingwood and Jack Burnham were run out off successive balls before Coughlin hit the winning boundary as rain poured down across the ground..
"It was a bit of a surprise that we managed to get out there, the outfield was really wet," Collingwood told BBC Newcastle.
"But, when you've got a win under your belt, your first win, you don't mind. I thought the guys showed a lot of character.
"Things hadn't been going easy in T20 this year, but to stand up and chase that total down after what was pretty much a Luke Ronchi blitz in the first innings, was a great effort."
Despite the victory, Durham remain bottom of North Group, with Leicestershire dropping two places to sixth.
Sussex and Middlesex managed to make a prompt start to their match in Hove, but were eventually beaten by the rain which swept across the country.
Middlesex were chasing a fourth T20 victory of the season and reached 136-6 from 17.4 overs when the players were forced off for the second time.
Captain Brendon McCullum smashed a quickfire 26 early on, but he fell to Chris Jordan and some impressive fast bowling from Jofra Archer (2-17) kept the visitors in check.
The worsening conditions meant the Sharks reply never got under way with the match abandoned just after 21:30 GMT.
The games between Lancashire and Nottinghamshire, Derbyshire and Northants, Birmingham and Yorkshire, and Glamorgan and Surrey were all called off before a ball could be bowled.
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Durham secured their first T20 Blast win in seven attempts as they beat Leicestershire in a frantic seven-overs-a side contest.