French striker Giroud broke the deadlock in the 39th minute when he swept home a Mesut Ozil corner.
The points were then sealed one minute from time when Rosicky's strike was deflected into the net by Everton centre-back Phil Jagielka.
Here, Sports Mole looks back over the contest to determine if the home side were worthy winners, or whether the scoreline was harsh on the visitors.
On target: 4
On target: 2
Was the result fair?
The statistics may show that Everton had more of the possession, but it is what you do with it that really counts. The visitors forced David Ospina to make one save of note, but that was about it. While way short of their best, Arsenal were both more clincial and efficient than their visitors and for that reason, they deserved the points.
Arsene Wenger would have demanded a positive response to the debacle of Wednesday night, but it never really materialised. For most parts his side were too pedestrian in possession, with very little movement off the ball.
It goes to show just how reliant the Gunners are on Alexis Sanchez for dynamism. The Chilean was off form today - as he has been in other recent outings - and it hampers his side badly. With Theo Walcott and Danny Welbeck among the substitutes and Alex Oxlade-Chamberlain playing from deep, it meant that there was very little speed in the final third.
The pluses for Wenger will have been Giroud getting his name on the scoresheet, particularly after he missed a host of glaring chances against Monaco in the Champions League. Also, new signing Gabriel made his full Premier League debut and, although there was a couple of nervy moments early on, he grew into the game and produced a solid display.
For want of a better word, watching Everton today was quite boring. Yes, they had plenty of possession, but there was far too much sideways passing between the halfway line and the Arsenal box.
One feature of their play last year was the freedom given to the full-backs to get forward and express themselves. Luke Garbutt did okay on the left, but Seamus Coleman on the other flank looked half the player that he was 12 months ago. Whether it is a change in tactics from Roberto Martinez or poor from the Irishman, the Toffees have lost a big threat from last year.
On the plus side, Romelu Lukaku was lively and although he did not score, he caused problems, which has not always been the case this season.
Sports Mole's man of the match
Olivier Giroud: There was no standout player from either side, but Giroud edges it. His confidence must have been hit in the wake of the Monaco game, but his response was good. His finish for the goal was instant and he spearheaded the Arsenal attack well.
Everton's centre-back pairing can share this award for their parts in the two Arsenal goals. John Stones allowed Giroud to run unopposed for the opener and then Jagielka was far too deep when Rosicky took aim, meaning that the deflection left Everton goalkeeper Tim Howard helpless.
It was a rather flat encounter, which meant that it was a routine 90 minutes for referee Andre Marriner. He brandished two yellow cards, both of which were warranted.
Arsenal: It is a London derby for the Gunners on Wednesday evening when they make the short trip across the capital to take on Queens Park Rangers at Loftus Road.
Everton: Meanwhile, on the same night, Everton are on the road once again - this time at Stoke City.