Hill struck in the 75th minute but Harry Redknapp’s men rarely showed the adventure expected of a side touted as favourites for promotion.
Yet it was good enough to secure a third league win from four matches this term.
Joey Barton, who endured persistent taunts from the Leeds crowd, had a goal wrongly disallowed for offside in the ninth minute.
The former Manchester City, Newcastle and Marseille midfielder ventured into the penalty box and expertly stroked the ball past former QPR goalkeeper Paddy Kenny from Gary O’Neill’s astute pass, only for the referee’s assistant to raise his flag for offside.
Televised replays showed Leeds defenders Lee Peltier and Scott Wootton playing Barton onside despite half-hearted protests from the visitors at the time.
Leeds had a penalty claim of their own rejected seven minutes after the break when Ross McCormack’s driven cross struck defender Danny Simpson’s arm just inside the box.
Rangers purged the memory of their home Capital One Cup defeat by Swindon in midweek but they had plenty of defending to do, with Leeds’ top scorer Ross McCormack looking to celebrate his new four-year contract and a recall to the Scotland squad.
Goalkeeper Robert Green dealt capably with a series of Leeds corners but Rangers broke out of their shell as the interval approached.
Kenny did well to pluck the ball off Barton’s head inside the six yard box and was then relieved to see Shaun Wright-Phillips drag a shot tamely wide after the lively winger had escaped Luke Murphy’s attention.
There were few moments to savour in a tepid first half but soon after the break Charlie Austin forced Kenny into a blocking save at the foot of a post and then drove close from an acute angle.
Leeds manager Brian McDermott made a double substitution in the 57th minute, replacing strikers Luke Varney and Noel Hunt with Matt Smith and Dominic Poleon – and Smith narrowly failed to get his head to Luke Murphy’s corner immediately after leaving the bench.
Rangers winger Junior Hoilett made his return from a hamstring injury, replacing O’Neill in the 64th minute. Alejandro Faurlin then took over from Karl Henry after 72 minutes and Rangers took the lead three minutes later.
Leeds made a hash of goalmouth attempts to clear Barton’s free-kick from the left and left-back Hill was first to react, lashing the ball into the net from a few yards out.
Leeds had gained seven of their previous eight points after falling behind and although Rodolph Austin rattled the bar with a 30-yard drive in time added on, Rangers clung on to capture the points.