Gibbs opened the scoring during the first leg of the sides’ UEFA Champions League qualifying play-off in Istanbul at the start of the second half with his first goal in the competition.
Theo Walcott was released inside the area by Ramsey, he rolled a low cross to the back post and Gibbs was on hand to fire into the roof of the net.
Ramsey added a second for the visitors just after the hour, the Wales international scoring a fine individual goal from the edge of the penalty area.
Giroud added the third from the penalty spot to put Arsenal on the verge of reaching the group stages.
Arsene Wenger was forced into one change from the opening day of the Premier League season with injury to Alex Oxlade-Chamberlain ruling him out of the trip to Turkey.
Santi Cazorla replaced the England international with Lukas Podolski having to settle for a place on the bench.
Fenerbahce gave starts to new signings Bruno Alves and Michal Kadlec, while another new arrival Emmanuel Emenike was named as a substitute.
Having dominated the opening exchanges, Arsenal tested goalkeeper Volkan Demirel for the first time after 19 minutes.
Cazorla created the first opportunity for Walcott, his through-ball nearly finding the forward but Demirel slid in well to intercept.
The loose ball fell to Tomas Rosicky and he delivered a perfect cross for Giroud but the striker’s header was just wide of goal.
Fenerbahce were struggling to get into the game and failed to test Wojciech Szczesny in the opening half.
Wenger's early season injury problems continued just after the half hour when Laurent Koscielny was forced off with a head injury.
The French defender clashed with Pierre Webo and had to be replaced by Carl Jenkinson.
The hosts began the second half at a much higher tempo but fell behind six minutes in as Gibbs scored a crucial away goal.
Just after the hour, Arsenal doubled their lead thanks to Ramsey.
He received a pass from Jack Wilshere 35 yards from goal and moved at pace towards the Fenerbahce defence. As he reached the penalty area, Ramsey fired a low shot into the bottom corner, although Demirel should have done better.
The victory was secured with 14 minutes to play as Kadlec brought down Walcott inside the penalty area. Giroud made no mistake from the spot, sending Demirel the wrong way to complete a perfect night for Arsenal.
