City looked as though they were cruising to victory when a deflected Aaron Mooy effort and a Stefan Mauk goal gave them a 2-0 lead at the break.
But a determined Jets outfit turned things around after the interval. Ben Kantarovski pulled one back, before Milos Trifunovic drew the visitors level with a penalty.
The Serbian striker then headed in the winner from a set piece to complete a remarkable comeback win for Newcastle.
In an open start to the contest, both sides had early chances, but it was Melbourne who took the lead on 12 minutes after Mooy received the ball outside the area and fired in a shot that deflected past the returning Mark Birighitti in the Jets goal.
Newcastle had an excellent opportunity to hit back in the 20th minute when a corner found its way to Nigel Boogaard, whose free header went wide.
The visitors were punished further less than two minutes later as City extended their lead with a sublime team effort.
An excellent combination move on the edge of the box allowed David Williams to slip Mauk through and the midfielder did enough to force the ball beyond Birighitti and make it 2-0.
The Jets began the second-half strongly and just before the hour they pulled a goal back.
Mateo Poljak lifted a ball back across goal from the right and Kantarovski rose highest to head in from close range.
City responded immediately, after a long ball from the back released Bruno Fornaroli inside the area.
The Uruguayan did well to jink past his defender and get a shot off, but the effort was saved by Birighitti and the follow-up shot blazed over the bar.
On 72 minutes, Newcastle had a golden chance to pull level from the spot after substitute Wade Dekker was ajudged to have handled in the box.
Trifunovic comfortably slotted the penalty into the top corner to make it 2-2.
Buoyed by the equaliser, the Jets then came agonisingly close to taking the lead minutes later.
A delightful cross from David Carney found Trifunovic all alone in the box, but his header was saved by Thomas Sorensen before the City defenders somehow scrambled the ball clear.
The hosts' defence could do little to stop the Serbian striker soon after though.
Again it was Carney with the delivery, this from a free-kick, and Trifunovic headed in from close range to complete the turnaround.
The 3-2 victory lifts Newcastle to the top of the A-League before the round's other fixtures take place.
The loss is City's first in 11 home games, but the three points to Newcastle ensures they have equalled their win tally from last season after just four matches.
Coach Scott Miller said he was delighted to find a win off the back of two goals from his Serbian striker.
"I think he might be the buy of the season," he said.
"Every performance is giving us a great framework to continue our development as a team and rebuilding of the club."
City boss John Van 't Schip, who said the referee denied Corey Gameiro a "clear" penalty at 2-1 up, saw his team abandon their passing game after their dominant opening stanza and revert to playing long balls.
"We should have indeed made the score bigger than the 2-0," he said.
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