Pep Guardiola's charges, who had lost their previous two matches, made the perfect start when Phil Foden bundled in a cut-back from Matheus Nunes in the first minute of the game.
The home side doubled their lead after 25 minutes when defender Josko Gvardiol pounced the quickest to a loose ball in the box.
The visitors looked out of the contest at halftime but made a remarkable fightback after the interval when Dominic Calvert-Lewin pulled one back.
Leeds were then awarded a penalty when Gvardiol fouled Calvert-Lewin and Lukas Nmecha equalised in the 68th minute from the rebound after Gianluigi Donnarumma had saved the spot-kick.
City threw everything at the visitors and just as it seemed as Leeds had done enough to earn a point, Foden stepped up to complete his brace in the first minute of stoppage-time.
The win moves Man City to within four points of Premier League leaders Arsenal, who travel to third-placed Chelsea on Sunday.
It was a fourth consecutive defeat for Leeds, who remain remain in the relegation zone having collected 11 points in 13 matches.
Manchester City secured a second successive Premier League victory at home against Sunderland, Chelsea were held scoreless away to Bournemouth while Tottenham managed to beat Brentford.
Manchester City manager Pep Guardiola insists his side are moving in the right direction as they continue to evolve following an off-season of significant rebuilding.
Enzo Maresca admitted that Cole Palmer's long-awaited return was "the only good news" on a punishing Wednesday night for Chelsea, who were beaten 3-1 by a sharp Leeds United side at Elland Road.
Leeds United hauled themselves out of the relegation zone with an energised 3-1 victory over Chelsea on Wednesday, a result that relieved growing pressure on manager Daniel Farke.
