The hosts trailed to Taylor Regan's 37th-minute opener at AAMI Park, putting them on course to surpass the sorry streak of 19 matches without tasting victory that they share with the defunct New Zealand Knights.
However, Iain Ramsay and Mate Dugandzic scored in quick succession at the end of the first half to turn the game on its head, and the latter then added a second in the 68th minute.
That allowed Melbourne to celebrate a long-awaited victory and a first in John van 't Schip's second spell at the club, after the coach returned in place of the sacked John Aloisi last month.
Orlando Engelaar, having recovered from a broken leg suffered in pre-season, was introduced from the bench for his debut prior to Dugandzic capitalising on an exquisite throughball from Ramsay to score the Heart's third.
In Friday's other game, league leaders Brisbane Roar overcame the first-half dismissal of striker Besart Berisha to claim a 0-0 draw at Perth Glory.
The Albanian international, who is infamously remembered for an apparent dive to win a late penalty in the 2012 grand final against Perth, was lucky not to have been sent off after just 20 seconds when he jumped into Thwaite's ribs during an aerial contest.
Berisha's exit sparked heated scenes on the pitch, with Glory goalkeeper Danny Vukovic and Roar midfielder Liam Miller booked for their involvement in the scuffle.
