After being under the cosh for the first half, Marc Nygaard set up Inigo Idiakez to nod Rangers ahead.
Nygaard's penalty, after Paul Furlong was fouled by Shaun Newton, doubled QPR's lead and the Dane sealed the points with a 30-yard volley.
Ian Hume scored a late consolation for Leicester from the spot after Danny Cullip had been penalised for handball.
QPR manager John Gregory:
"In the first half we were horrendous. We were tired and lethargic and allowed Leicester to dominate the early exchanges.
"I would have expected Leicester to do much better with all the possession they had but they never really punished us.
"It was a fine win in the end and we now have three matches at home. If we manage to win those it might just get us across the finishing line."
Leicester: Henderson, Maybury, Kenton, Kisnorbo, Johansson, Newton, Jarrett (Wesolowski 76), Hughes (Tiatto 76), Yeates, Hume, Horsfield (Hammond 62).
Subs Not Used: Logan, Stearman.
Booked: Johansson, Hume.
Goals: Hume 90 pen.
QPR: Camp, Mancienne, Cullip, Stewart, Bignot, Ainsworth (Smith 88), Bolder, Idiakez (Lomas 63), Cook, Furlong (Ray Jones 84), Nygaard.
Subs Not Used: Cole, Kanyuka.
Booked: Stewart, Cullip, Furlong, Ray Jones.
Goals: Idiakez 47, Nygaard 51 pen, 68.
Referee: N Swarbrick (Lancashire).