RADEK Cerny's heroics couldn't stop battling QPR going down at Anfield.
The Rangers keeper produced a string fine saves to keep Liverpool at bay, but was helpless to stop Luis Suarez's winner early into the second-half.
The hosts flew out of the box, with Maxi and Martin Skrtel seeing early efforts blocked, while Suarez headed straight at Cerny when he was found completely unmarked six yards out.
Rangers had a lucky escape minutes later when Maxi played in Suarez, but the striker skewed his shot wide.
Cerny came to QPR's rescue on the half-hour mark when Dirk Kuyt's ball sliced through the visitors' backline, but Maxi was denied by the Czech keeper.
However, Rangers' resolve was eventually broken two minutes after the break.
Charlie Adam sent in a ball from the right, and Suarez got between Luke Young and Armand Traore to head Liverpool in front.
QPR were dealt another blow soon after when Anton Ferdinand limed out of the action, with Luke Young reverting to centre-back.
Cerny's heroics and the post kept out Maxi as Liverpool pushed for a killer second, while the keeper denied the Argentine again minutes later.
Neil Warnock threw on DJ Campbell as they searched for an equaliser, with Bothroyd stabbing an effort over the bar.
But despite a period of late pressure, they were unable to find it as they returned to W12 empty handed.
QPR: Cerny, Young, Gabbidon, Ferdinand (Orr 50), Traore, Smith (Campbell 66), Barton, Faurlin, Wright-Phillips, Mackie (Hill 78), Bothroyd.
Subs (not used): Putnins, Connolly, Derry, Buzsaky.
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