The pace bowler made light work of the 19 needed from the final over to earn the Rapids a second win in the space of 24 hours.
Finch set the tone by depositing the first ball from Fisher over the mid on boundary and then the fourth and fifth deliveries also disappeared over the ropes to spark wild scenes of celebration.
The 22-year-old ended with 30 not out from just 10 deliveries as the Rapids squeezed over the finishing line with one ball to spare.
He came in at 145-7 in the 18th over and struck all but one of the remaining 31 runs required to pull off a remarkable victory against the odds.
It means Yorkshire are still searching for their first win of the season and in all they have gone 11 matches since their last triumph in all cricket last August.
David Wiese showed all his experience with the ball on his Vikings debut and looked to have seen his side to victory.
The 38-year-old, who has been playing for Kolkata Knight Riders in the IPL, scored a quickfire 15 but it was his bowling which bamboozled the home side.
He mixed up his pace and line to great effect to finish with 3-18 from four overs – but then came Finch’s dramatic assault.
The Vikings were restricted to 175-9 after opting to bat with Pat Brown picking up three wickets in his final two overs to finish with 3-28.
Spin accounted for 11 of the Rapids overs with Club Captain Brett D’Oliveira picking up two more wickets after his 3-28 against the Steelbacks while Michael Bracewell and Usama Mir bowled tightly.
The Rapids made one change from the side which had launched the competition with victory away to Northamptonshire Steelbacks, with Dillon Pennington replacing fellow pace bowler Mitchell Stanley.
The Vikings gave a debut to former Sussex all-rounder David Wiese who has been playing in the IPL for Kolkata Knight Riders.
Yorkshire captain Shan Masood opted to bat but his side were on the back foot after Dillon Pennington struck in each of his opening two overs.
Dawid Malan steered the first delivery he faced straight to keeper Ben Cox and Adam Lyth’s mistimed pull lobbed straight into the hands of mid on.
It brought back memories of Pennington’s explosive 4-0 new ball burst in a Blast game at Headingley two years ago.
Jonny Tattersall and Masood set about trying to rebuild the innings during a partnership of 47 but both fell in successive overs from Rapids captain Brett D’Oliveira.
Tattersall was pouched at short fine leg sweeping at the leg spinner who then saw Masood hole out off a slog sweep to deep mid wicket.
The spin combination of Usama Mir, Michael Bracewell and D’Oliveira kept the scoring rate in check during the middle overs until Jordan Thompson struck three successive sixes off the latter.
He raced to 36 off just 18 balls before aiming another big hit at Bracewell and skying a simple chance to backward point.
Wiese announced himself in impressive fashion with a flat six off Adam Finch over extra cover but then the former Sussex player and Ben Mike both lofted Pat Brown straight to deep mid wicket in the same over.
Some big blows from Matthew Revis – including successive sixes off Finch - and Dom Bess gave the Vikings innings late impetus with 31 runs gleamed from the 19th and 20th overs.
Revis top-scored with 42 from 31 balls before he was run out in the final over from Brown who then bowled Jafer Chohan after he backed away to the final delivery.
The Vikings carried their momentum into the start of the Rapids innings with a double breakthrough from Matthew Fisher.
D’Oliveira mistimed a pull and perished just inside the boundary and Bracewell chased a wide delivery and was caught behind.
Adam Hose, who top-scored with 61 against the Steelbacks, was quickly into his stride and struck Thompson for three fours in the final over of the powerplay.
Jack Haynes also looked in good touch and on drove and off drove Revis to the boundary as the 50 stand was completed in 32 balls.
Thompson broke the partnership after it added 78 when he held onto a fine return catch away to left to account for Hose.
Haynes went in the next over from Jafer Chohan when he failed to clear deep mid wicket.
Ben Cox clubbed his way to 26 off 14 balls before he turned Revis straight to deep square leg and Ed Pollock aimed a legside blow and was bowled by Wiese.
When Kashif Ali was caught at cover off Thompson, it looked all over for the Rapids but then Finch went into overdrive to secure victory.