Elkay Gundogan put the homeowner ahead with a shot from the edge of the penalty area in the 19th minute, and eight minutes later added the second goal in the same way.
The German player missed a penalty 6 minutes from time, after his colleague Erling Haaland left it for him.
In the next minute, substitute Rodrigo pulled one back for Leeds, setting the stage for a thrilling end, but City held on to their advantage.
Manchester City have taken their tally to 82 points from 34 games, four points ahead of Arsenal, who have played the same number of games.
City have spoiled the debut of new Leeds manager Sam Allardyce, who took over after Javi Garcia was sacked last week.
Leeds remained in 17th place with 30 points, ahead of Nottingham Forest, who host bottom club Southampton on Monday, on goal difference.
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