Liverpool dominated from the outset and coasted to victory even without Rodgers, who was unwell and too sick to be on the touchline.
Assistant manager Colin Pascoe and coach Mike Marsh deputised at Loftus Road, which witnessed two Suarez goals inside the opening 16 minutes.
Daniel Agger nodded the visitors further ahead in the 28th minute and, despite an improved second-half performance from the hosts, Liverpool were never in any real danger of losing grip of proceedings.
The defeat leaves QPR bottom and eight points adrift of safety heading into a tough January, which starts with a trip to rivals Chelsea.
Suarez netted his first of the afternoon in the 10th minute. The Uruguayan collected a pass from Jordan Henderson and ghosted past Clint Hill before striking low past Julio Cesar.
Liverpool doubled their advantage with Suarez again both scorer and creator, bursting along the byline from the right to produce a cross that Nedum Onuoha could only block straight into the striker's path, allowing him to net again.
Liverpool did not have to wait long for their third goal, as Steven Gerrard collected a short corner and sent in a right-wing cross for Agger to nod home.
Jordan Henderson was inches away from Liverpool's fourth with a drilled 20-yard strike, before a Gerrard effort was cleared off the line by a combination of Armand Traore and Ryan Nelsen.
Rangers talisman Adel Taarabt struck a number of shots from distance but to no avail.