Alexis 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 to script in the early exchanges, as the hosts completely dominated possession amid some dogged defending from struggling QPR.
The opening goal came eight minutes before half-time when Alexis, who had missed an early penalty, headed home.
Arsenal were 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 visitors late hope when hammering home from the penalty spot.
A leveller was not forthcoming, though, as Rangers' unhappy run of not picking up a single point on the road this term 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 were restricted to speculative efforts from distance, most notably from Giroud and Alexis.
Even when the QPR defence was breached, Green was there to make the save, epitomised 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 Alexis to send a diving header into the net.
To their credit, QPR responded solidly, with Steven Caulker directing a header just wide before the break.
And their hopes were given a significant boost in the 54th minute when Giroud's loss of temper left Arsenal with 10 men.
The hosts made light of their one-man disadvantage, though, as they initially kept the ball impressively.
They were aided, too, by the brilliance of Alexis 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 materialise, as Austin notched his 12th league goal of the season after Mathieu Debuchy was harshly adjudged to have fouled Junior Hoilett.
Mauricio Isla went closest to equalising 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.
