Hobart posted a competitive total of 5-182 at Marvel Stadium despite the league's leading run-scorer Ben McDermott failing for three, victim to a sensational catch from Zak Evans.
The Renegades threatened to get over the line and clamber out of last spot on the table, thanks chiefly to Finch's 75.
Tuesday night's penultimate over, delivered by Tom Rogers and costing just three runs, proved the game changer.
Rogers removed Finch and Jonathan Merlo with consecutive deliveries after both batters tried to clear the rope, leaving Sam Harper and Will Sutherland needing 15 runs from the final over.
It proved too tough as Melbourne finished on 6-176 after a Riley Meredith full toss in the final over was judged a legal delivery rather than a no-ball on height.
The result means Hobart join Perth Scorchers, Sydney Sixers, Sydney Thunder and Adelaide Strikers in advancing to the play-offs, although the jostling for ladder positions will continue with the last three games of the regular season on Wednesday.
"I just looked to take the pace off and hoped it worked," Rogers told Fox Sports, when asked about claiming the key wicket of Finch.
"I've unfortunately been on the receiving end of some of Marshy (Shaun Marsh) and Finchy's finest work.
"We just had to stay patient. Wickets were going to fall eventually.
"To be able to go into that last game without having to win is a big relief.
"Also going into that game and heading into finals after a real close, hard hit-out is good. It's a pretty good practice run."
A fired-up Matthew Wade (48), D'Arcy Short (37) and Tim David (46 not out) put their side on track for victory, stepping up after McDermott's dismissal in the third over.
Finch and Shaun Marsh (51) shared a classy 101-run partnership in response, with Melbourne's equation reduced to 45 runs from 31 deliveries when the latter departed.
Finch, who could have been run out on 25, crashed the next delivery he faced for a sweetly timed six while marching to his highest score of the season.
But Hobart, forced to 'host' the clash at the Renegades' home deck after the league rejigged its fixtures in an attempt to limit the impact of COVID-19 on the competition, closed out the game after tidy overs from Sandeep Lamichhane, Meredith and Rogers.
Unmukt Chand, who made history by becoming the first Indian to play in the BBL, scored six on debut for the Renegades.
Wade was struck on the body by a return throw while top scoring for his side, prompting a fierce response that was picked up by the stump microphone.
"I'm f***ing in my crease," Wade said.
"The pitch is there, how hard is it to get it near the stumps?"
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