Playing in just his fourth game, the 18-year-old smacked 18 from nine balls in a crucial 27-run partnership from just 14 balls with Derek de Boorder (25).
Needing 23 runs from the final two overs to chase down Northern's impressive 308 for eight, Finnie hit the first two balls of Jono Boult's penultimate over to the fence to put the pressure on the bowling side.
After taking 14 off Boult's over, Otago needed nine runs from the final six deliveries to bounce back from its 97-run shellacking by Auckland on Friday.
Finnie hit the first ball of the over for a single, before Scott Kuggeleijn bowled an ill-timed no ball, off which de Boorder took a single.
The right-handed Finnie sealed the Volts' third win in its first four games of the Ford Trophy by hitting a four off the next delivery and a couple off the third ball of the over.
Otago's thrilling chase would not have been possible without the platform openers Aaron Redmond (64) and Sam Wells (93) expertly set up.
The pair put on 134 runs for the first wicket, batting 27.5 overs in the process.
When Redmond was dismissed by Ish Sodhi, Michael Bracewell came in and upped the ante by bludgeoning 83 from 48 balls, including 10 fours and four sixes.
After Wells departed with the score 247 for three, the required run rate exceeded 10 an over with new batsmen Nathan McCullum and de Boorder at the crease.
McCullum could only muster five, while Iain Robertson (seven) did not fare much better.
However, Finnie joined the classy de Boorder at the crease and the pair capped an excellent run chase to ensure the Volts did not lose consecutive games to start the new year.
Earlier, Daryl Mitchell struck an unbeaten century at just over a run a ball to guide his side to 308 for eight.
He came to the wicket with his side 70 for two after 11.3 overs, and quickly set about his work. The No 4 was involved in three 50-run partnerships in his almost three-hour innings.
Mitchell's 100 runs came from 98 balls and included eight fours and three sixes.
Dean Brownlie scored 66 runs at the top of the order, while Mitchell Santner hit a breezy 45 at a run a ball.
Jacob Duffy was the chief destroyer for the Volts, snaring four for 64 from his 10 overs.
Neil Wagner, who was overlooked by the Black Caps in favour of Doug Bracewell for the second test match against Sri Lanka, conceded 72 runs from his 10 overs and failed to pick up a wicket.
Otago A had a thrilling win against Northern A on Saturday and a comprehensive seven-wicket loss to Auckland A at Lincoln yesterday.
In Saturday's win, Ryan Duffy hit a fine century and Sean Eathorne struck a quick-fire 82 from 59 balls to help Otago chase down Northern's 292 on the last ball of the innings.
By Robert van Royen.
