The ex-Tottenham winger opened the deadlock with a deft lob over City keeper Paul Jones.
Routledge doubled the Championship club's lead seven minutes later when he rifled home Patrick Agyemang's pass.
City were reduced to ten men when Scott Golbourne handled Kaspars Gorkss' header on the line. Routledge made no mistake from the spot to complete the first treble of his career.
QPR heaped more misery on the home side when Allesandro Pellicori blasted home from close range before Hogan Ephraim added a fifth.
The Championship side carved out the first chance of the match, Rowan Vine watching in horror as his looping header bounced off the top of the crossbar and out to safety.
Winger Routledge then brought a smart save out of Paul Jones after bursting into the box to collect Vine's defence-splitting pass.
City started to muscle their way into the game and Liam Sercombe made an electric run into the box, only to be hustled out of his shooting stride by Mikele Leigertwood.
Marcus Stewart produced a clever turn before shooting straight at Radek Cerny as the home side grew in belief as the half progressed.
A scintillating passage of play ended with Ryan Harley crossing for Stewart to send an angled header into the grateful arms of Cerny.
The giant Czech keeper was then called upon again to foil Bertie Cozic after the Frenchman let rip from 25 yards.
The game came to life after the break with Routledge making the breakthrough with a superb finish.
He wasted little time in doubling the visitor's advantage with an angled drive past Paul Jones.
Routledge sealed the win from the penalty spot for his hat-trick before a clinical finish from Pellicori put the gloss on the victory. A 25-year pearler from Ephraim poured further salt into City wounds.