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.
The Jets, missing four first-team regulars due to international duty and a further two long-term injury victims, struggled to mount a notable foray forward in the first half-hour, but moved ahead when centre-back Regan was left unmarked from a corner to head home his first A-League goal.
It did not take long for Melbourne to respond, Ramsay levelling with a deflected volley before Dugandzic stooped low and converted a cross from left-back Aziz Behich.
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.
Berisha was given his marching orders in the 30th minute after receiving his second yellow card of the evening for a crude challenge on Glory defender Michael Thwaite.
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.
But his night ended on the half-hour mark when he raised a knee into Thwaite and was shown red, much to the delight of the Perth crowd.
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.
Despite having an extra man, the hosts were unable to take advantage, as Brisbane stretched their lead at the summit to five points.
