Alvaro Negredo pulled one back late on for City, but it was not enough to make a difference.
Despite City's bright start, it was the Gunners who took the lead inside nine minutes, as Walcott latched on to Ramsey's diagonal pass into the box before lifting the ball over Joe Hart and into the far corner.
They nearly drew level in the 21st minute when Wojciech Szczesny made a hash of catching a high cross into the box before spilling it to Negredo on his left. The striker's shot was looping towards the far corner but Kieran Gibbs came to the rescue by heading it off the line.
However, Arsenal came inches away from doubling their advantage through Bacary Sagna after he collected possession in the City half before slamming a shot from range, but Hart managed to push it onto the post.
City made the better start after the break and came close to a leveller through Negredo after nice play from David Silva and Jesus Navas, but his low effort from the right flew wide across the face of goal.
However, Arsenal managed to double their lead in the 59th minute when Walcott slotted a pass through to Ramsey, who rounded Hart on the right before passing the ball into the back of the net.
The Gunners instantly added a third just three minutes later when Walcott's 40-yard pass from the right was brought down on the edge of the box by Giroud before he lobbed the ball over Hart to make it 3-0.
Giroud thought that he had made it 4-0 in the 73rd minute after slamming home a wonderful finish from a brilliant Walcott through-ball from the right, but he was flagged marginally offside.
That's as good as it for City, though, as the two sides now prepare for the upcoming Premier League season.