Gibbs opened the scoring in the first leg of the play-off in Turkey in the 51st minute, netting from close range after good work by Theo Walcott and Ramsey.
Ramsey doubled the lead 11 minutes later with a low 20-yard effort which was too hot for Fenerbahce goalkeeper Volkan Demirel to handle.
And Giroud rounded off the success from the penalty spot in the 77th minute, after Walcott had been brought down.
Although the Gunners lost defender Laurent Koscielny to a facial injury sustained in the first half, it was just the morale-boosting performance Arsene Wenger was seeking.
Wenger had challenged his players to rise above the negativity which followed the 3-1 home defeat to Aston Villa in their Barclays Premier League opener on Saturday and deliver the required performance at the Sukru Saracoglu Stadium.
And despite being without several injured players in Istanbul, he got it as Arsenal limited the home side to few openings and silenced the home support.
Fenerbahce, whose place in the European competition is still subject to a hearing by the Court of Arbitration for Sport after they were banned for match-fixing, created little.
In contrast, the Gunners were lively throughout - despite being without several first-teamers because of injury.
And the north London side were rewarded with a result which makes them overwhelming favourites to be in next Thursday's draw for the group stages.
Source: PA
Source: PA