Warwicks travelled to the St Lawrence Ground in Canterbury to play Kent in a Group C game and won the toss and opted to field first.
Chris Woakes struck first for Warwicks taking the wicket of Key before Denly and van Jaarsveld put on 113 for the second wicket.
Denly was finally out for 80, bowled by Imran Tahir before fit again Darren Maddy took the wickets of Mahmood and Jones.
Stevens had hit five sixes in an innings of 55 before he was caught by Westwood off the bowling of Neil Carter to see Kent close their innings on 246-6.
Jon Trott and Neil Carter opened for Warwicks and Carter hit one four before he was out with the score on twelve. Ian Bell joined Trott at the crease and the duo added 126 runs before Trott was out for 54 from 73 balls.
Ian Bell had hit two sixes and ten fours but he was the third Warwicks player out when he caught and bowled by van Jaarsveld for 88.
Darren Maddy was next in but only last two balls before trapped lbw by Nel.
Jim Troughton put in a superb innings, making 56 from 51 with one six and six fours and Ian Westwood chipped in with an unbeaten 28 as Warwicks won by six wickets with 13 balls to spare.
On Tuesday, Warwicks are in County Championship action against Hampshire at Edgbaston. Play is due to start at 11am and live commentary can be heard by clicking on the link on BBC Coventry & Warwickshire.