Sullivan, the 21-year-old leg-spinner playing only his second first-team game, performed the feat with the opening four deliveries of his spell as Derbyshire totalled 108-6 after being sent into bat.
Fraine then struck a career-best, unbeaten 69, reaching his half-century from just 19 balls, as Yorkshire ran out eight-wicket winners with eight balls to spare, climbing to second in Group B, two points behind leaders Glamorgan, who they face in their final group game in Cardiff on Thursday.
The top-three from the two groups go into the knockout stages, which begin on Saturday, with the group winners advancing straight into the semi-finals, while the second and third-placed teams contest eliminator fixtures.
Sullivan, who took 1-43 on debut against Nottinghamshire at York on Friday, said: “It’s been a good few days for me, and I’ve really enjoyed it.
“Once you’ve played your first first-team game, and you do alright, it gives you the confidence and belief that you can perform at that level, and, to be fair, a lot of the time I think that’s the battle.
“Anything can happen when a game gets shortened like this one did, and I knew that their batsmen were going to come hard at me and it was just a case of trying to be one step ahead of them.
“Luckily it paid off on this occasion, and I’m just happy that I was able to contribute.”
Fraine, who has several times shown his ball-striking prowess in T20 cricket, made what appeared a tough target seem like a walk in Queen’s Park itself as he raced to a half-century that included four fours and four sixes.
The 25-year-old, who set the tone by striking three boundaries in the first four balls of Yorkshire’s reply, landed one straight blow on the pavilion balcony, but he was not fussy about where he whacked it in a brilliant display.
“To help get the lads over the line was something that I’ve been striving to do all year, and it was just a great team performance,” said Fraine, who faced 32 deliveries in total.
“I’d not had a big score, as such, in this competition, but I’d been feeling really good and threatening this and I’ve just been backing myself really and believing that this was going to come.
“This time it clicked, and I managed to get a few out of the middle, and now we move on to a big game on Thursday.
“It’s great to be in the hunt for qualification with a young side; we’ve had the backing of the coaches to go and express ourselves, and we’ll go down to Glamorgan with our tails up and hopefully get a win down there.”
After play at Queen’s Park was delayed by six hours, following torrential morning and afternoon rain, Sullivan sent Derbyshire spinning from 49-1 to 50-4 in the still soggy conditions when the action finally got going at 5pm.
First, Tom Wood slog-swept to deep mid-wicket, where Fraine took the catch in front of the popular bank, where a decent number of Yorkshire supporters were rewarded for their patience – not least in putting up with chanting from home fans who were suitably well-oiled after such a long wait.
With the batsmen crossing while Fraine took the catch, opener Harry Came was back on strike for Sullivan’s next ball, which he lofted up to long-on for a single.
Sullivan – showing no sign of nerves and plenty of guile – then had Brooke Guest stumped as he came down the pitch and Alex Hughes caught by Matthew Revis as he swept to square-leg, both men falling for golden ducks.
Sullivan captured a fourth wicket in his second over – bowlers were permitted only two overs maximum in a game that should have been 50 overs per team – when Came skied high into the offside, captain Gary Ballance judging a good catch running out towards cover as his young bowler finished with the remarkable figures of 4-11.
Fynn Hudson-Prentice top-scored for the hosts with an unbeaten 38 from 17 balls with four sixes, the other wickets going to George Hill, who bowled Mitchell Wagstaff around his legs, and Matthew Waite, who had Mattie McKiernan brilliantly caught by Fraine at deep mid-wicket.
Thanks to Fraine, Yorkshire were home and hosed at the halfway stage of the chase, at which point they were 72-1 after five overs. Revis was the man dismissed, skying to mid-on, with Will Luxton joining Fraine in a second-wicket stand that had realised 64 when Luxton departed in the seventh over, caught in the deep on the leg-side off the right-arm medium-fast bowler George Scrimshaw, having contributed 20 from just 11 balls with two fours and a six.
Ballance emerged and was soon clubbing Hudson-Prentice for a handsome straight six as his team completed an emphatic triumph.
This has been a bogey ground for Yorkshire in recent times; they had lost on six of their previous seven visits in white-ball cricket going back to 2014.
Yesterday it was just a boggy ground, one where it looked as though we were not going to see any play until Sullivan and Fraine made one grateful that we did in the end.