Giroud opened the scoring in the fifth minute with a first-time finish past Sunderland goalkeeper Vito Mannone.
Giroud scores twice on Gunners return
The Frenchman struck again just after the half hour mark when he intercepted a sloppy back pass from Santiago Vergini intended for Mannone and slotted the ball underneath Arsenal’s former backup ‘keeper.
Rosicky scored just before the half-time whistle with a chip after a string of one-time passes from Arsenal, and Koscielny netted with a free header in the 57th to wrap it up.
Sunderland substitute Emanuele Giaccherini scored a late goal with nine minutes to go, but it was nothing more than a consolation for the Black Cats.
The result leaves Arsenal in second place with 59 points, one behind Chelsea, which won earlier in the day.