The Friends Life t20 runners-up reached their target of 151 with three balls to spare at the Wanderers in Johannesburg,as they defeated the Sri Lanka Premier League winners.
Deprived of England stars Jonny Bairstow and Tim Bresnan against the tournament debutants, Yorkshire restricted Uva Next to 150/7 in their 20 overs after Andrew Gale (pictured) decided to bowl first.
Dilshan Munaweera, one of three Uva players fresh from the World T20, opened the batting and made 22 with Bhanuka Rajapaksa (24) before veteran Shivnarine Chanderpaul added 27 and Thilina Kandamby boosted the total with an unbeaten 29.
Former Australian Test opener Phil Jaques made 32 in Yorkshire's reply before Dilshan Munaweera removed him and Gary Ballance next ball to leave Carnegie at 68/4 after eight overs.
Adil Rashid and David Miller added 25 before the South African was forced to retire hurt after being struck in the face by an Umar Gul bouncer.
But after the removal of Dan Hodgson (18) with 18 needed off the final 10 balls, Miller hit Munaweera for four, six and four in consecutive balls, leaving the English side needing three to win off the final over.
Miller hit the winning runs to finish unbeaten on 39 with Rashid 36 not out.
If Yorkshire beat Trinidad & Tobago tomorrow, they will top qualifying tournament Pool 2 and join Sydney Sixers, Chennai Super Kings, Mumbai Indians and Lions in Group B of the main tournament.
In the day's other match New Zealand's Auckland Aces ended the world record 25 match unbeaten run of Pakistan side Sialkot Stallions at the Wanderers.
Kyle Mills grabbed two wickets for the loss of just six runs to help limit the Stallions to a total of 130-9 off their 20 overs; and Auckland then reached 136-4 after just 17.1 overs as Martin Guptill top-scored with 40 while Anaru Kitchen had a late blast of 33 off 21 balls.