Two minutes of play early in the second half typified the game.
Brisbane winger Lachlan Maranta was somehow held up over the line by Billy Slater when he looked a certainty to score.
A Broncos error on the next play handed Melbourne the ball and, in the next set of six, Storm lock Ryan Hinchcliffe ran on to a Finch pass and managed to get the ball down at the other end.
While Finch pulled the strings in attack, Storm winger Sisa Waqa was just as impressive.
Waqa saved two tries in the first half, racing across field to nail Brisbane winger Lachlan Maranta with the tryline beckoning and also foiled a likely Josh Hoffman try from a Peter Wallace kick.
He then scored midway through the second to cap a flawless performance.
Finch, 31, took over the Storm playmaking duties following the season-ending hip injury to Gareth Widdop.
His work in attack, as well as some try-saving tackles in defence, will give the Storm confidence they have the right man to steer them through their finals campaign.
While Melbourne continued to frustrate the Broncos, the scoreboard kept ticking over in the second half through Hinchcliffe, Waqa and a double by centre Maurice Blair.
Compounding the misery for the Broncos, centre Jack Reed suffered a shoulder injury by hitting the signage behind the dead ball area as he attempted to bat the ball back into play.
Brisbane coach Anthony Griffin said Reed had re-injured the shoulder he had previously had reconstructed and would go for scans.
"It doesn't look good," Griffin said.
"He's a pretty tough customer, Jack, so when he comes off you know he's hurt."
Griffin wasn't making excuses for his team, saying they lacked patience and execution.
"We needed to be a lot more dogmatic about what we were trying to do with the ball.
"We had multiple sets on our own line but we just didn't have the execution or intent to really earn some points."
The Broncos must win six of their remaining eight games to make the finals.
Melbourne coach Craig Bellamy was delighted with the way his team responded from their 22-4 loss to Wests Tigers, which was their second successive defeat.
"We were really disappointed with our performance last week and the boys came out with a really good attitude tonight," Bellamy said.
"Some of our tryline defence tonight was outstanding - there was a couple of real close shaves there."
As well as his team scoring six tries, he felt that stopping so many of the Broncos' likely four-pointers hurt the visitors.
"That can cause some frustration when you get that close and you can't get over the line.
"They probably played good enough footy to get over the line but some last-ditch defence saved it for us."
- AAP
