Reigning champions United's victory left City knowing that if they lost away to Arsenal later on Sunday they would be eight points behind their cross-town foes with just six games of the season left.
Goals from Wayne Rooney and Paul Scholes saw United to victory but Rooney's opener was controversial.
The England striker gave United a 15th minute lead from the penalty spot after QPR captain Shaun Derry was sent off for bringing down Ashley Young inside the area and was deemed by referee Lee Mason to have denied a clear goalscoring chance.
Mason's decision was harsh on Rangers, who are still above the relegation zone on goal difference. Derry's contact appeared minimal and Young looked to be offside.
But even though QPR keeper Paddy Kenny correctly dived to his right, Rooney's well-struck penalty gave him his 30th goal of the season and left the visitors, managed by former United striker Mark Hughes, with a mountain to climb.
Kenny saved from Rafael da Silva and Danny Welbeck to keep QPR in the game until the 68th minute when he was beaten by a 20-yard shot from veteran midfielder Scholes, recalled from retirement this season by United manager Sir Alex Ferguson.