After a couple of half-chances, Heidar Helguson should have hit the target on 13 minutes when Routledge, who returned to the QPR side on loan this week, picked him out at the second attempt but the Icelandic striker headed into the ground and over the bar.
Coventry were content to absorb the early pressure with King their most potent outlet on the break, but the opening goal came from a more direct route.
Westwood kicked long, and after a King flick-on, the ball was set back to McSheffrey who delivered across the edge of the six-yard box for the former Watford striker to steer in.
Freddy Eastwood could have extended Coventry's lead after a quick break but scuffed his shot when well placed.
Just as it looked like the visitors would head into the break in front, the dangerous Taarabt collected Paddy Kenny's deep free-kick, cut inside two defenders and curled his shot into the far corner for a goal deadly in its precision.
QPR full-back Clint Hill then shot narrowly over as Coventry desperately held on before half-time.
The second half began in similarly open fashion with Alejandro Faurlin hitting a post from a curling free-kick before King's set-piece engineered a fine double save from Kenny, who also denied substitute Lukas Jutkiewicz from the follow up.
Warnock, who had also added Pascal Chimbonda and Ishmael Miller to his squad this week, then introduced the latter and the former West Brom striker again brought out the best in Westwood as he tipped round the post.
But the keeper was caught out by Taarabt's crafted pass for Routledge's goal, as the Moroccan forward sliced open the Coventry defence with a pass from the left touchline with the outside of his right boot.
Routledge marked a fine debut by controlling the ball superbly and coolly placing home to give Rangers a comfortable cushion at the top of the Championship.
QPR boss Neil Warnock: "Either goal was fit to win any game.
"And you won't see a better ball for the second anywhere in the world.
"There's baggage with him and he'll make some mistakes. He was bombed out by the previous two managers here and rightly so.
"But it couldn't be better for me at this time of my career and I think I'm the right manager for him.
"My son bought his mate today just to see Taarabt. He thinks he should be playing for Real Madrid. And he could, but first of all he's got to help me."
Coventry boss Aidy Boothroyd: "The difference was Taarabt.
"He's an exceptional player, I was tempted to run on and kick him myself at one stage. He was absolutely terrific.
"How do you stop him? I'm not really bothered now, we've played them twice.
"He didn't do much at our place and I don't think he did too much today, but then suddenly he hits a 40-yard pass with the outside of his foot and you have to admire it.
"With that talent you have to ask why is he here?"
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