Premier League: QPR 0 Liverpool 3
Suarez struck twice in the opening 20 minutes before Daniel Agger headed home from a corner in a dominant first half for Liverpool.
Liverpool manager Brendan Rodgers missed the trip through illness so Colin Pascoe took charge, making three changes from Boxing Day's defeat to Stoke, while the most notable change in five for QPR saw Rob Green dropped after his clanger in defeat to West Brom and replaced by Julio Cesar.
The away side started brightly and Uruguayan Suarez forced an early save from Cesar with a fierce drive that the Brazilian parried toward Allen, but the Welshman scuffed his volley when following in.
Suarez was not to be denied in the 10th minute as he picked the ball up 30 yards out and drove toward goal, leaving Clint Hill in his wake before calmly slotting home from inside the area.
Six minutes later, Liverpool and Suarez had a second as Stewart Downing released his star team-mate down the right, who lashed the ball across the box towards Sterling.
Nedum Onuoha's last-ditch block ran back to the former Ajax man, who swept home from six yards.
The hosts had a strong shout for a handball turned down as Jordan Henderson elbowed Adel Taarabt's free-kick clear before Stephane Mbia's bicycle kick flew over the bar.
Harry Redknapp's side were split open again soon after, as Martin Skrtel's long ball from the back released Glen Johnson.
The right-back stormed forward before scuffing across goal, beating Cesar, but Ryan Nelsen hacked away with Suarez awaiting to tap in.
The Hoops were undone again before the half-hour mark as Downing's quick corner found Gerrard, who swung in a cross that an unmarked Agger nodded back across goal and past Cesar at the far post, which had been vacated by Shaun Wright-Phillips.
Suarez was running rampant and arrived in the box before cutting back to Gerrard. The Liverpool skipper smashed an effort that Armand Traore partially blocked before Hill scrambled off the line.
QPR were being overrun in midfield but showed some spark late in the first half, as Taarabt smashed a couple of efforts from range, one of which required a simple save from Reina.
Redknapp responded at the break by replacing Djibril Cisse with Shaun Derry, but little changed as Sterling burst in from the left before feeding Suarez, whose stinging left-footed effort drew a flying save from Cesar.
Liverpool's tempo dropped in the second period, with Reina being called into action by Taarabt and Mbia, while substitute Esteban Granero scuffed a low free kick wide with his first touch for the home side.
QPR ran out of steam late on, with only a tame Mbia free-kick to show for their efforts as the game meandered to a close, Liverpool climbing one place to ninth and ensuring the beleaguered Hoops will end 2012 rooted to the foot of the table.