Cazorla opened the scoring with a well-taken finish in the first half which comfortably beat Caballero, before Ramsey doubled the lead before-half time.
City rallied for a brief spell after the break following the introduction of David Silva, but Giroud sealed the win shortly after the hour mark with his deflected effort.
Injuries and late returns from World Cup commitments combined to leave City without seven of the XI that sealed a second Premier League title in three seasons with a final-day win over West Ham in May.
Arsenal were also missing several of their main men, with Per Mertesacker, Mesut Ozil and Lukas Podolski, who helped Germany triumph in Brazil, unavailable.
It was Arsenal who acquitted themselves to far greater effect in the first half, as new signing Alexis Sanchez, Jack Wilshere, Ramsey and Cazorla all buzzed inventively behind Yaya Sanogo.
Arsenal dominated the opening exchanges but clear chances evaded them and it required a last-ditch challenge from Mathieu Debuchy to prevent Samir Nasri from breaking the deadlock against his former club in the 14th minute.
That chance did nothing to turn the tide for the Premier League champions, for whom Yaya Toure and Fernando were being badly overrun in midfield and the opening goal stemmed from Arsenal exploiting generous amounts of space in front of City's back four.
Wilshere went to ground under a 21st-minute challenge from Dedryck Boyata but the ball broke to Cazorla on the edge of the box, who shifted the ball onto his left foot to finish clinically.
Arsenal debutant Calum Chambers then made a timely intervention to deny Nasri with the goal gaping on the end of Aleksandar Kolarov's 39th-minute cross, but Wenger's men grabbed a deserved second three minutes before the interval.
City were torn apart on the break and Ramsey collected Sanogo's pass, evaded Matija Nastasic with a cute touch and slotted home.
Sanchez had chances to extend the advantage in stoppage time and a ragged City backline were relieved to hear referee Michael Oliver's half-time whistle.
Silva's impact was instant as City resumed on the front foot and Stevan Jovetic headed against the post from Jesus Navas's 51st-minute centre before blazing the rebound off target.
Jovetic then received Kolarov's cutback to draw a sharp near-post stop from Wojciech Szczesny - in what was the Arsenal goalkeeper's first save of note.
But Arsenal sealed victory in style after the hour, as substitute Giroud lashed a dipping 25-yard strike beyond Caballero's despairing dive, to lay down an early marker ahead of the new season.