The win snapped a run of five successive losses by the Renegades at Etihad Stadium and kept the Thunder winless after the first two games of their BBL title defence.
After losing the toss, the Renegades made 7-179 from their 20 overs, with the Thunder all out for 131 in the 20th over.
Finch, who admitted his place in the Australian limited-overs side was under pressure after a lean run, belted 63 runs off 37 balls, his knock including three fours and four sixes.
The Renegades skipper smashed successive sixes off the first two balls of Fawad Ahmed’s third over but was caught on the rope as he tried for three in a row.
But by then, he’d successfully set the tone for the Renegades, looking to reach the finals for just the second time in six attempts.
South Australian Callum Ferguson was next top scorer for the Renegades with 38 from 32 balls, with Dwayne Bravo putting in a cameo of 24 runs from 16 balls.
Andre Russell, who also affected an athletic run-out, was the leading wicket taker for the Thunder but was expensive with figures of 2-42.
The Thunder’s run chase began disastrously when Kurtis Patterson spooned the first ball of the innings from Tom Cooper to point where Chris Tremain took the easiest of catches.
And when Eoin Morgan fell for two in the second over, caught by Marcus Harris off Tremain’s bowling, the Thunder were on the ropes.
Ben Rohrer and Ryan Gibson came together to steady the ship with a 70-run partnership but boundaries were few and far between as the required run rate ballooned.
Evergreen spinner Brad Hogg was given a hero’s welcome by the 23,015-strong crowd when he came on for his first spell since crossing from the Perth Scorchers.
He claimed his first wicket for the Renegades in his third over when Gibson, who top scored with 39 off 37, skied a return catch off a cheeky bouncer.
And when the 45-year-old bowled Rohrer for 32 just four balls later, his cult hero status was confirmed and the fight drained from the visitors.
Hogg finished with 2-22 from four overs, with fellow Renegades debutant Sunil Narine snaring 2-32 and Bravo claiming 2-28.
“It was a pretty polished performance in the end,” Finch said.
“There are some areas we can improve but to get the first-up win (was great).
“We’ve had some pretty good sides in the past but I think now we’ve got guys who are in form and that kind of confidence flows on.”
Thunder coach Paddy Upton urged patience as his side, currently without star skipper Shane Watson, builds into the season.
“It’s never nice to lose but it’s been important for us to be clear on what our campaign is going to look like,” he said.
“How we learn, how we grow and how we build the kind of cricket we want to play.
“If we stay on the right path and do the right things hopefully we’ll start clicking more and more.”
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