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.
Arsenal started brightly, with Walcott seeing the night's first chance - a low driven shot - blocked by defender Bruno Alves in the fourth minute.
There were concerns for Jack Wilshere in the 15th minute when he went down under a challenge from Mehmet Topal - but the midfielder was able to run off an apparent foot injury.
Four minutes later Santi Cazorla, back in the starting line-up with Alex Oxlade-Chamberlain facing a stint on the sidelines, almost found Walcott. Demirel was alert to the danger, clearing with his feet, before Giroud headed wide.
Walcott and Giroud combined to set up Ramsey for a shot which the Wales international midfielder dragged wide in the 25th minute.
Five minutes later and the Gunners were forced into a change when Koscielny was caught in the face by Pierre Webo's high boot as the pair jostled for possession. The Fenerbahce striker got a yellow card, while a bloodied Koscielny was replaced by Carl Jenkinson.
The Gunners continued to play the tidier football but their pleasing approach-play did not lead to many chances in the remainder of the first half. Giroud's shot over the bar from 12 yards shortly before the break was their only noteworthy effort.
Fenerbahce's first decent opening came in the fourth minute of the second half, when former Liverpool striker Dirk Kuyt fired narrowly over after Wilshere had been dispossessed.
Arsenal were ahead within two minutes. Ramsey found Walcott with an excellent ball and the England man crossed for Gibbs to convert from barely four yards out.
It was nearly 2-0 soon after when Walcott was denied from six yards by Demirel - before play swung to the other end and Wojciech Szczesny was out bravely to deny Webo.
Webo skewed a shot wide in the 59th minute, then Wilshere saw a 62nd-minute effort parried to safety by Demirel.
However, the goalkeeper was not so convincing two minutes later. Demirel appeared to be keeping out Ramsey's low shot from 20 yards out - but the ball eluded his grasp and squeezed into the corner of the net.
Moussa Sow shot over for Fenerbahce in the 71st minute and former Chelsea and Liverpool midfielder Raul Meireles fired wide in the 75th but Wenger's men were well in control of the tie.
And it was 3-0 when Walcott won a penalty in the 76th minute when he was bundled over by Michal Kadlec and Giroud converted emphatically, sending Demirel the wrong way.
Substitute Emmanuel Emenike came close to a consolation goal in the 90th minute - but Szczesny was equal to his shot.