Kieran Gibbs, Aaron Ramsey and Olivier Giroud scored in the second half to face down a noisy crowd in Istanbul, plus hostile media and Gunners fans at home after an off-season of failed transfer deals for the cash-rich club.
Under-fire manager Arsene Wenger now looks all but certain to lead his team into the Champions League group stage for the 16th straight season after the return match next Tuesday.
"We could have been a little bit shaky after what happened on Saturday," said Wenger, whose team lost 3-1 at home to Aston Villa in the Premier League opener. "So for us it was very important to start on the front foot. That was vital tonight for us and we did it well."
Away teams dominated Wednesday, as Swiss champion Basel won 4-2 in Bulgaria against Ludogorets Razgrad and Austria Vienna won 2-0 at Dinamo Zagreb.
Schalke drew 1-1 at home to PAOK Thessaloniki and it was also 1-1 between Steaua Bucharest and Legia Warsaw.
Arsenal's dominating performance almost made irrelevant the pre-match speculation of chaos awaiting UEFA if Fenerbahce advanced while appealing a two-year Champions League expulsion over a 2011 Turkish match-fixing case.
That appeal verdict from the Court of Arbitration for Sport is scheduled next Wednesday, when Arsenal will likely be thinking of the next day's 32-team group stage draw in Monaco.
Arsenal's neat passing impressed from the start though Giroud and Ramsey spurned its only half-chances before the break. The visitors lost defender Laurent Koscielny to a nasty-looking kick in the face from Fenerbahce forward Pierre Webo, who was booked.
Koscielny was left bleeding from an open wound and was taken to a hospital for a checkup after the match.
"It's very deep, it's very, very deep," Wenger said of Koscielny's wound.
Yet Wenger's players surged on and defender Gibbs struck in the 51st when connecting with Theo Walcott's slick pass across the goalmouth.
Ramsey's 22-meter (yard) shot in the 64th slipped under Fenerbahce goalkeeper Volkan Demirel's diving grasp, and Walcott earned a penalty by drawing a foul from defender Michal Kadlec.
Giroud fired the 77th-minute spot kick decisively to Demirel's right, and Arsenal goalkeeper Wojciech Szczesny preserved the clean sheet with excellent late saves.
PAOK thrived on its reprieve by UEFA, having already lost this month to Metalist Kharkiv which was then disqualified for fixing a Ukrainian league match in 2008.
The Greek side earned a 1-1 draw in Germany when Miroslav Stoch levelled in the 73rd to cancel Peru forward Jefferson Farfan's perfectly placed low shot for Schalke in the 32nd.
Egyptian midfielder Mohamed Salah starred for Basel, scoring twice and earning a late penalty.
Ludogorets, which was playing semi-professional, third-tier football three years ago, levelled Salah's 12th-minute opener when Brazilian midfielder Marcelinho scored in the 23rd. The Bulgarians led in the 50th through Ivan Stoyanov's low shot after two teammates failed to connect with a cross.
Salah levelled in the 59th and, five minutes later, Giovanni Sio made it 3-2 for Basel when exploiting more gaps in the home side's defence. Junior Caicara was sent off for fouling Salah in the box, and defender Fabian Schaer converted the 84th-minute penalty.
Dinamo also finished with 10 men. Forward Ante Rukavina was shown a red card in the 71st, three minutes after Marin Leovac put Austria Vienna ahead. Marko Stankovic made it 2-0 to the Austrians in the 75th.