Sterling had endured widespread derision after England manager Roy Hodgson revealed the Reds star had told him he was too tired to play in last weekend's Euro 2016 qualifier against Estonia.
With Hodgson watching from the stands, Sterling won a free-kick which led to Hoops defender Richard Dunne turning into the net for Liverpool's second-half opener.
That set the stage for an astonishing finale as Eduardo Vargas equalised with three minutes left, only for Philippe Coutinho to restore Liverpool's lead in the 90th minute.
Vargas levelled again in stoppage time, but Liverpool still emerged with three points thanks to Sterling, whose cross was deflected into his own net by QPR defender Steven Caulker with virtually the last kick of the match.
"Sterling was outstanding in his running and in his desire, but QPR certainly did not deserve to lose," said Rodgers. "We were lucky to get the win, but we showed tremendous character."
Liverpool were well below their best, but somehow they were able to secure a morale-boosting victory ahead of Wednesday's crunch Champions League clash with Real Madrid.
Van Gaal eager for first away success
Manchester United manager Louis van Gaal believes his team's away form is better than their record suggests as they look for a first win on the road at West Bromwich Albion on Monday.
Van Gaal, whose side have recovered from a poor start to the season to climb to fifth place in the table ahead of Sunday's matches, claims the points they have amassed on their travels are not a true reflection of the way they have performed.
"We played very good at Leicester City,” said the Dutchman. "But the other matches, we played two draws and we could have also won those matches."
Chelsea's whole new ball game pleases Mourinho
Jose Mourinho said Chelsea's 2-1 win away to Crystal Palace showed they have the character to maintain their Premier League title challenge.
However, the Portuguese boss said it was too soon to talk of the Premier League trophy returning to Stamford Bridge.
"If we were in another league, another profile of league, I would say immediately 'yes, we will do it'," said Mourinho. "In the Premier League, we can do it. But it's hard to say we will do it.”
Arsenal slip-ups a worry for Wenger
Arsenal manager Arsene Wenger has said his players do not currently possess the ruthlessness to challenge Chelsea and Manchester City for the Premier League title.
The Gunners were held to a 2-2 draw at home to Hull City on Saturday and required a stoppage-time equaliser from England forward Danny Welbeck to avoid losing the game.
"Of course I cannot say it's not a concern," said the Frenchman. "It's a big concern because the other teams win their games and we don't."
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