The Storm looked to have the win in the bag when they fought back from a Shark-dominated first half with second half tries from Jesse Bromwich and Will Chambers to lead 12-8 with only 14 minutes left on the clock.
The Sharks, who finished bottom of the league two years ago in the wake of a doping scandal, would not let their dreams of breaking the longest title drought in the league die, however, and Andrew Fifita put them back in front 10 minutes from time.
"I can't believe it, I thought we were gone when Chambers scored, they hardly concede anything," said emotional Sharks hooker Michael Ennis, who was playing in the final match of his career.
"Melbourne are champions, they just cover and cover and cover. It's unbelievable!"
James Maloney opened the scoring with a penalty in the seventh minute and it was the Sharks five-eighth's brilliant break that almost put Luke Lewis in for the first try.
From the five-metre scrum that followed, Sharks skipper Paul Gallen picked up the ball from the second row and sent Ben Barba scuttling down the blindside to score.
The Sharks dominated possession and territory for the remaining 25 minutes of the half but the vaunted Storm defence somehow managed to keep them out.
Ten minutes after the break, though, Storm forward Bromwich forced his way over the line and when winger Chambers danced through the defence to add another converted try 15 minutes later, Melbourne were ahead.
The Sharks kept their heads up, however, and in the 70th minute prop Fifita charged onto a pass from Ennis and took four tacklers over the line with him to score the winning try.
The Storm threw everything they had at the Sharks in the dying minutes, switching play from one side of the field to the other, and Chambers could have set up Cooper Cronk to score had he spotted him free inside after collecting a chip-and-chase.
"With 15 to go it maybe looked like we had the game," Storm captain Cameron Smith said.
"But they're a special footy team, they just kept plugging away, they didn't panic and they deserved the win tonight.
"Everyone that follows rugby league must be happy for the Sharks, they've waited a long time for this."
