Giroud turned in a volley from Mesut Ozil’s corner in the 39th minute of the Emirates Stadium encounter before Rosicky made sure of the points with a late deflected effort.
Romelu Lukaku and Aaron Lennon went closest for Everton, who have now won just one of their last 11 league matches, but both were denied by good saves from David Ospina.
While the result continues Everton’s poor run of form, it is the ideal response from Arsenal following their disappointing Champions League defeat to Monaco in midweek.
Per Mertesacker paid the price for that performance as he was replaced by Gabriel Paulista, who made his first Premier League start.
The Brazilian grew in confidence as the game progressed but looked nervy at times in the first half and was nearly caught out in the 17th minute when he hesitated on the ball.
Lukaku took advantage to nick it away from him, but, fortunately for Gabriel, his keeper was more decisive and did well to clear his lines.
It took Arsenal until the half-hour mark to create their first chance as Alexis Sanchez clipped in a cross from the left to Giroud, who got ahead of Phil Jagielka but headed wide.
Gabriel then made a superbly-timed sliding tackle to halt Lukaku as he charged into the box.
The deadlock was broken not long afterwards as Ozil whipped in a corner from the right and Giroud swept a volley into the far corner.
The goal appeared to give Arsenal a lift and Santi Cazorla forced Tim Howard to tip a shot over the bar as the half drew to a close.
But Everton made a positive start to the second half and Gareth Barry sent an acrobatic volley over the bar.
Giroud saw a close-range effort blocked by Jagielka at the other end before Cazorla fired over the bar after a clever one-two with Giroud.
Ospina then produced a fantastic reflex save to turn over Lukaku’s near-post drive and got down well to save Aaron Lennon’s low shot from Seamus Coleman’s cut back.
There was a lengthy stoppage after a clash of heads between Giroud and team-mate Francis Coquelin left the midfielder with blood pouring from his nose.
Coquelin eventually had to be taken off late in the game but Arsenal wrapped up the win when Rosicky’s shot deflected off Jagielka and flew past Howard.
Arsenal: Ospina (7), Bellerin (6), Gabriel (6), Gibbs (6), Koscielny (5), Cazorla (6), Coquelin (6), Oxlade-Chamberlain (6), Ozil (6), Sanchez (6), Giroud (7).
Subs: Chambers (5), Rosicky (7), Welbeck (5).
Everton: Howard (5), Coleman (6), Stones (5), Jagielka (5), Garbutt (5), Barry (6), Besic (5), McCarthy (6), Barkley (5), Mirallas (5), Lukaku (5).
Subs: Gibson (5), Lennon (6), Naismith (5).
Man of the match: David Ospina.