The Frenchman has admitted his future is in doubt following Alexandre Lacazette's arrival in a club record £52.6million deal.
But Giroud, 30, grabbed the opener at ANZ Stadium as he started in the same team as Lacazette.
The pair struggled to linked up together although Lacazette did have a hand in the build up to Giroud's goal.
He sent through Nacho Monreal and the Spaniard's cross was slammed into the top corner by Giroud.
Monreal also set up the second goal with a long pass that led to Aaron Ramsey lobbing the ball past Vedran Janjetovic.
Mo Elneny wrapped up three goals in 11 first-half minutes when his effort deflected in off Brendan Hamill.
West Sydney pulled one back in the second half when Ramsey's pass back was picked up by Janjetovic.
Steven Lustica's first effort from six yards was blocked but he slammed in the rebound.