City played the majority of the game with 10 players, after Phil Foden was red carded for kicking out at United captain Bruno Fernandes, but they pinched victory at Old Trafford as Erlng Haaland stabbed home on the hour mark.
Haaland was initially flagged offside, with the goal ruled out, but referee Michael Oliver was advised to allow the goal following a VAR check.
The Norwegian was just onside as he tapped home Josko Gvardiol's deflected cross, but the United boss Michael Carrick was angered that Enzo Fernandez's attempt to score was not deemed as interfering with play.
Fernandez was clearly offside, and refereeing body Pro Ref admitted the goal should have been ruled out, with Rooney left furious over the error.
"I'm just so sick of VAR. It's horrendous and it's taking all of the emotion out of the game.
"To not recognise that Enzo Fernandez is offside is baffling. Anyone who has played football at any level sees that as offside."