The North London side took a 3-0 lead into the game, but quickly extended that to a four-goal advantage, before the Welshman added his third of the tie in the second half.
Arsene Wenger's side will learn who they will face in the next stage of the tournament when the group stage draw takes place later this week.
Sports Mole looks back at tonight's match to analyse the result.
On target 7
On target 3
Was the result fair?
Absolutely. Arsenal caused problems for Fenerbahce all night. They could have sat back, having headed into tonight with a 3-0 lead, but Arsene Wenger picked an attacking side who looked dangerous every time they found their way into the Fenerbahce box. There could have been no other result.
The Gunners were superb this evening. Theo Walcott, Carl Jenkinson and Aaron Ramsey controlled the midfield, while Santi Cazorla's pace tore holes through the visitors' defence. Arsenal always looked like scoring, and probably could have had a few more goals, but for some good saves from Volkan Demirel. At the back, Per Mertesacker and Nacho Monreal made some crucial blocks for the home side, while Wojciech Szczesny made some stunning stops just before half time. All in all, a brilliant all-round performance by Arsenal.
The Turkish side showed some promise up front in the first half, as Moussa Sow and Emmanuel Emenike worked well together. However, it was the defenders that let Fenerbahce down this evening. They gave the likes of Olivier Giroud, Ramsey and Lukas Podolski far too much room, allowing Arsenal to create countless chances.
Sports Mole's man of the match
Aaron Ramsey: Who else could it have been? After setting up one goal and scoring another in the first leg, the Wales international continued to test Fenerbahce this evening. As well as his two brilliant finishes for tonight's goals, the midfielder also looked to set-up his teammates throughout the match, including one flick back for Giroud, which could have given him his hat-trick had he taken it himself. A booking for coming back onto the pitch after changing his boot without asking permission is the only blot on his performance this evening.
This has to go to Emenike for his awful attempt at winning a penalty in the second half. The Nigerian was chasing down a ball, when Szczesny came out to make the clearance. However, the striker tried his luck, diving before the Arsenal keeper was within three feet of him! Unfortunately for him, he did not fool the referee, as was swiftly booked for his theatrics.
Carlos Velasco Carbello had a very solid game tonight. Fenerbahce had a couple of penalty shouts, but the Spanish referee correctly waved both away. The Turkish side started getting a little physical towards the end as they tried to taunt the Arsenal players, but the official quickly stepped in to straighten things out while keeping 22 men on the pitch.
Arsenal: The Gunners will await Thursday's group stage draw before the small matter of the North London derby on Sunday, as Tottenham Hotspur head to the Emirates Stadium with a 100% record so far this season.
Fenerbahce: The Turkish side will hear tomorrow whether their two-year ban from European football will be upheld before facing Sivasspor in the Super Lig on Saturday.