Ilkay Gundogan needed just 19 minutes to get Manchester City off the mark with a well-taken passing shot into the bottom corner, before doubling his tally - and his side's lead - just eight minutes later from almost exactly the same position.
Leeds were much improved in the second period, though only Erling Haaland's uncharacteristically wayward shooting kept them from conceding a third, as the Norwegian hit the woodwork on several occasions.
Gundogan then missed the opportunity to score a first career hat-trick from the spot, before Rodrigo scored for the visitors 58 seconds later to set up a grand stand finish.