Haaland, who was born in Leeds when his father Alf-Inge represented the Whites, missed two good chances as the champion dominated the first half, but Rodri poked it ahead on the stroke of half-time.
The Norwegian striker was not to be denied after the break, tapping home after an error from home captain Liam Cooper before doubling up with a right-footed finish.
That meant Haaland reached 20 Premier League goals in record time on his 14th appearance, helping City move back into second place despite Pascal Struijk's consolation.
City almost went ahead inside 36 seconds, but Illan Meslier got a hand to Haaland's chipped effort after the striker was released Nathan Ake.
The Leeds goalkeeper made another one-on-one save from Haaland on the half-hour mark, before Jack Grealish missed two glaring opportunities to break the deadlock, the second a dire mishit from Rico Lewis' cutback.
However, the hosts' resistance was finally broken in first-half stoppage-time, with Rodri pouncing to bundle home when Meslier denied Riyad Mahrez after a flowing City move.
Leeds was master of its own downfall five minutes after the restart, as Cooper's sloppy pass allowed Grealish to steal possession and tee up Haaland for a simple finish.
Haaland had his second of the game just past the hour, Meslier failing to keep out his powerful right-foot shot.
Leeds cut the deficit 17 minutes from time when Struijk headed Sam Greenwood's corner beyond Ederson, though City managed to see out the win despite Joe Gelhardt threatening.
