Finch hit five sixes and three fours in his 46-ball knock and his dismissal left the Renegades needing 31 to win with three wickets in hand.
Finch batted at No.3 rather than opening after a knee injury to Glenn Maxwell forced a reshuffle of the top order.
Shahid Afridi took Finch's opening role, partnering Brad Hodge, and Maxwell came in at No.7.
Maxwell hurt his knee while unsuccessfully trying to stop a boundary during the Heat innings and hobbled through his knock of 10, before being caught at mid-off for Neser's third wicket.
The Heat were in massive trouble when they lost 4-10 to be 5-74 in the 12th over.
Leg-spinner Shahid Afridi took three wickets in four balls.
But Vettori and van der Merwe batted superbly in their 72-run stand to first stop the collapse and then compile a decent score.
Vettori had made 22 when he hit a Dirk Nannes delivery down the throat of deep square-leg.
But it was a no-ball because Melbourne had too many men outside the circle.
- AAP