Mahela Jayawardene showed his class at the crease as he made a half-century, but his teammates failed to support his efforts as only one other player reached double figures leaving the home side with 140 in their 20 overs.
Durham skipper Mark Stoneman got his team off to a fast start but he fell for 35. An innings of 43 from Phil Mustard kept the home side up with the rate, but Ajmal put the clamps on the batsman to guide his team to victory, despite the best efforts of Gordon Muchall (41*).
Birmingham Bears put the pressure on Durham at the top of the North division as they moved level on points with the Jets following their seven-wicket win over the Nottinghamshire Outlaws via the Duckworth-Lewis method.
Riki Wessels was the only bright spot of the Outlaws' innings as he made 39 from 19 deliveries, although James Taylor (23) and Darren Sammy (23) also contributed to push their side to 149-9 from their 20 overs.
However, Vince smashed five sixes and eight fours in his innings and was supported by Sean Ervine (37), who hit three sixes of his own, to reach the victory target with 13 balls to spare.
Michael Klinger responded with a century of his own as he blasted 10 fours and three sixes to reach the milestone, but only two other players made double figures to leave the hosts 20 runs short of their target.
Croft hit nine boundaries and two sixes, while James Faulkner also took the attack to the Falcons with an unbeaten knock of 47 off just 30 balls as the home side posted 201-3 in their 20 overs.
The match between the Northamptonshire Steelbacks and the Leicestershire Foxes ended without a result as rain ended the game in the home side's reply at Grace Road.
Results in full:
Glamorgan 191-3 beat Gloucestershire 172-6 by 19 runs
No result between Northamptonshire Steelbacks and Leicestershire Foxes
No result between Somerset and Surrey
Lancashire Lightning 201-3 beat Derbyshire Falcons 134 ao by 67 runs
Worcestershire Rapids 159-5 beat Durham Jets 156-7 by three runs
Hampshire Hawks 187-4 beat Kent Spitfires 183-5 by six wickets
Birmingham Bears 101-3 beat Nottinghamshire Outlaws 149-9 by seven wickets by D/L method
Essex Eagles 144-3 beat Sussex Sharks 140-9 by seven wickets