The midfielder, who will join the Los Angeles Galaxy at the end of the season, headed in Philippe Coutinho's corner in the 87th minute at Anfield.
Fifth-place Liverpool moved four points behind Manchester United with three matches remaining. United, in the fourth Champions League place, has played a game less.
QPR is second from bottom, seven points from safety with nine points to play for.
Leroy Fer had volleyed QPR level in the 73rd minute, canceling out Coutinho's first-half opener, but the visitors had Nedum Onuoha sent off 10 minutes later for fouling Jordon Ibe.
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