Philippe Coutinho scored the only goal of the game in the first half with a superb strike to give Carragher a winning send-off after an illustrious career at Anfield.
Carragher almost scored in the second half when his stunning long-range effort came back off the post before being replaced late on to a standing ovation from the home fans.
Liverpool started brightly and they felt they should have opened the scoring after just two minutes through Coutinho. The Brazilian got on the end of Stewart Downing’s corner at the far post and saw his header kicked off the line by Michael Harriman, although it appeared like the ball had crossed the line before the QPR defender’s intervention.
QPR had claims for a penalty waved away four minutes later when Loic Remy went down in the box under pressure from Lucas, but referee Martin Atkinson waved away their appeals.
Liverpool made the breakthrough on 23 minutes with a superb long-range strike from Coutinho. Debutant Jordon Ibe teed up Coutinho 25 yards from goal and he drilled in a brilliant low shot into the bottom corner past Robert Green.
Remy went close for QPR on 35 minutes when he produced a smart turn and shot from the edge of the box, with his left-footed effort whistling just wide of the target.
Liverpool continued to call the tune early in the second half and Jose Enrique saw his shot spilled by Robert Green, and seconds later Ibe cut inside from the left and hit in a shot which drifted just wide of the far post.
The home fans almost got the moment they came for just past the hour mark when Carragher picked up the ball from fully 30 yards and drilled a shot which smacked off the post with Green well-beaten.
Liverpool had further chances to score with Green saving well from Enrique and Downing seeing a shot cleared off the line by Armand Traore.
Carragher was then replaced by Sebastian Coates in the final five minutes to a standing ovation from the crowd as he bowed out in style.