Olivier Giroud's red card early in the second half set up a nervy finish, but goals from Alexis Sanchez and Tomas Rosicky secured the points for Arsene Wenger's side.
Sanchez scored a first-half opener and, after Giroud had been sent off following a moment of madness, then set up Tomas Rosicky for a second as Arsenal took the points at the Emirates Stadium.
The game went according to script in the early exchanges, as the host completely dominated possession amid some dogged defending from struggling QPR.
The opening goal came eight minutes before halftime when Alexis, who had missed an early penalty, headed home.
Arsenal was hampered after the restart by Giroud's ill-disciplined sending off after 53 minutes - the striker shown a straight red card for pushing his head into the face of Nedum Onuoha.
However, Rosicky - making his first Premier League start of the campaign as one of two changes for Arsenal - grabbed a second goal, only for Charlie Austin to give the visitor late hope when hammering home from the penalty spot.
A leveler was not forthcoming, though, as Rangers' unhappy run of not picking up a single point on the road this season continued.
It appeared they would be chasing the game as early as the eighth minute when former Arsenal man Armand Traore clumsily brought down Alexis in the area.
However, the Chilean's penalty was as poor as the challenge that earned it, his weak effort easily saved by Rob Green.
Arsenal continued to enjoy the better of proceedings as QPR, going back to the 3-5-2 formation unsuccessfully deployed earlier in the campaign, struggled to gain a foothold - although a back three featuring Rio Ferdinand, selected as one of five changes to Redknapp's XI, initially held firm.
Arsenal was restricted to speculative efforts from distance, most notably from Giroud and Alexis.
Even when the QPR defense was breached, Green was there to make the save, epitomized by a fine stop that kept out Giroud's 20-yard free-kick.
The opener finally did arrive in the 37th minute following a stunning Arsenal move. Danny Welbeck and Santi Cazorla combined before the latter laid a pass into Kieran Gibbs, whose superb cross allowed Sanchez to send a diving header into the net.
To its credit, QPR responded solidly, with Steven Caulker directing a header just wide before the break.
And its hopes were given a significant boost in the 54th minute when Giroud's loss of temper left Arsenal with 10 men.
The host made light of its one-man disadvantage, though, as it initially kept the ball impressively.
It was aided, too, by the brilliance of Sanchez as he created the second goal. Having picked up the ball inside his own area, the Chile international charged at the defence before laying the ball into Rosicky, whose deflected effort eluded Green.
Hopes of a comfortable evening from that point did not materialize, as Austin scored from the spot to notch his 12th league goal of the season after Mathieu Debuchy was harshly adjudged to have fouled Junior Hoilett.
Mauricio Isla went closest to equalizing when his header from Niko Kranjcar's corner was cleared off the line, while substitute Bobby Zamora almost connected with a ball that flashed across the box, but Arsenal held on.