On his starting debut, 18-year-old Ben Mazzeo scored the opener inside 12 minutes on Sunday to give the visitors the upper hand.
Their second goal coincided with Leckie's injection into the contest for his first ALM minutes since suffering a knee injury on Socceroos duty in September.
Sluggish in defence, the Jets struggled to make the best of opportunities. A missed free kick at close range midway through the second half was particularly costly.
The win, City's second of the season, hoists the struggling reigning premiers to seventh on the ladder. They are 2-1-1 since sacking coach Rado Vidosic two games into the season.
The Jets have lost back-to-back games against bottom four sides and sit ninth on the table.
Croatian winger Marin Jakolis had a hand in both of City's goals, first dispossessing Dane Ingham on the left wing and sending a long cross to Mazzeo, who headed the ball home.
Jakolis sparked the counter-attack for City's second goal when he stole possession from Phillip Cancar in the 58th minute.
The ball found Jamie MacLaren, who gave City a comfortable lead with his 148th ALM goal.
Playing on the right wing, Leckie was rusty on his injection into the contest in the 56th minute, fouling Reno Piscopo to give the Jets a chance to peg a goal back with a free kick.
Leckie came close to scoring with a header at close range but a third City goal was not to be.
Just as time was running out for the Jets to fight back, City goalkeeper Jamie Young received a yellow card for taking forward Archie Goodwin down on the right wing.
The shot-stopper faced a nervous wait as referee Shane Skinner ran a VAR check, before doubling down on his decision to award only a yellow card.
The Newcastle crowd booed Young with each touch of the ball thereafter, believing Goodwin had been fouled in a scoring opportunity.
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