A spectacular half-century from New Zealand international Luke Ronchi, hit off just 19 balls and including seven fours and three sixes, set the Foxes on their way.
And although Yorkshire took wickets regularly in the increasingly nervy reply, Colin Ackermann's unbeaten 58, made off 47 balls, saw the hosts cross the line in the final over.
Remarkably, it was their first victory at home in the competition this season, with their previous five wins all coming away from the Fischer County Ground. Their two remaining fixtures see them face Derbyshire at home next Thursday, before they travel to Trent Bridge to play Nottinghamshire the following evening.
Yorkshire, who arrived with their spirits - and chances of qualification - lifted by beating Roses rivals Lancashire at Headingley the previous evening, must now beat Northants at home on Thursday, their final group fixture, to have any chance of qualifying.
Yorkshire's total was based on an explosive 75 from Tom Kohler-Cadmore, the right-hander's best T20 score since joining them from Worcestershire, made from just 40 balls.
Placement and timing was the key, particularly through the offside, in a 30-ball half-century which included 10 fours. Two maximums followed, muscled through wide mid-wicket off Callum Parkinson, but the young left-arm spinner had his revenge when he deceived and bowled Kohler-Cadmore in the last of his four overs, his 14th wicket in the competition this season.
Their other major contribution was made by Shaun Marsh, whose half-century came seven balls more slowly, with four of his seven boundaries cut behind square on the off-side.
Having been 129 for three off 14 overs, a total of 200 looked well within their capabilities, but tight Foxes bowling, notably from Matt Pillans, and an unexpectedly effective four overs from occasional off-spinner Aadil Ali, took some of the pressure off the Leicestershire batsmen.
Ronchi's wonderfully clean hitting reduced it further - the highlight an upper-cut six high over third man off David Willey - and although he slog-swept Azeem Rafiq to into the hands of Jack Leaning on 57, Ackermann's sensible innings ensured Leicestershire got over the line.