David Miller (39 not out) and Adil Rashid (36 not out) were the top scorers for the winners.
Opener Phil Jaques gave the chase right impetus after the early departure of skipper Andre Gale and Joe Root.
But after racing his way to 32 off 21 balls, Jaques offered a return catch to Dilshan Munaweera.
Miller, who came back after retiring hurt, along with Rashid, then finished the job for Yorkshire, giving them hope of qualifying for the main tournament that kicks off on Saturday.
Miller struck four fours and a six during his 29-ball knock while Rashid found the fence two times and cleared it once during his 30-ball innings.
Opting to bat, a disciplined bowling effort earlier helped Yorkshire restrict Uva Next to 150 for seven.
Steven Patterson and Moin Ashraf picked up two wickets apiece while there was one each for Ryan Sidebottom, Azeem Rafiq and Rashid.
Thilina Kandamby top-scored for Uva with an unbeaten 29 even as his other batters failed to convert the starts into big scores.
Kandambi aside, three other batsmen got to the 20s, while Andrew McDonald made 17.
For Yorkshire conditions here were pretty similar to the ones they are used to at home. Cloudy skies, following rain earlier in the day, and a green pitch made it an all-too familiar territory for the English side.
It prompted Gale to have a bowl and though Yorkshire were far from their best, they did not let Uva run away with the game.
Opener Munaweera, one of three Uva players fresh from the World T20 campaign, lifted Patterson for six in his first over but, after another three fours, tried to play the same lofted drive without moving his feet and found a diving David Miller at cover.
Just like Munaweera, batsmen that followed, also perished after getting starts, affecting their chances of posting a commanding total.