The visitors were stretched by Slaven Bilic's well-drilled outfit, who almost took the lead right from kick-off at the Ataturk Olympic Stadium through a long-range effort by former Chelsea striker Demba Ba.
However, Arsenal held out and - but for some poor finishing by Olivier Giroud - might have scored an away goal, with substitute Alex Oxlade-Chamberlain seeing his shot touched onto the post in the closing stages.
Besiktas may technically have been playing away, with their Vodafone Arena home being redeveloped, but you would not have known it given the ferocity of the Turkish fans, with deafening whistling greeting every touch by a player in a red shirt.
Arsenal broke early down the right through Alexis Sanchez, and his cross was hooked goalwards by Giroud as he fell, but blocked inside the six-yard box the France striker could not hook the rebound in.
Serbian referee Milorad Mazic remained calm despite continued calls for free kicks and yellow cards as Arsenal defended resolutely.
Bilic, the Croatian former West Ham and Everton defender, continued to prowl his technical area - across some eight lanes of running track around the perimeter of a pitch more suited to field events than a qualifier for top-level European competition.
Besiktas should have scored just before half-time. A long ball forward down the left was missed by Calum Chambers, the teenager signed from Southampton who had been playing very well, and Ba skipped clear towards the penalty area. However, the striker dragged his low shot wide of the left post.
Arsenal could have stolen the lead themselves, as a one-two with Jack Wilshere put Giroud free in a central position, but he could not make contact and the ball rolled through to the goalkeeper. Nacho Monreal was tripped by Veli Kavlak to concede a free kick just outside the Besiktas penalty area. Santi Cazorla struck a low effort which the keeper gathered at the second attempt.
Arsenal were reduced to 10 men on 80 minutes after Ramsey collected a second yellow card for pulling back Oguzhan Ozyakup and will now miss the return leg. The visitors almost snatched victory when Oxlade-Chamberlain, on for Sanchez, drilled the ball goalwards, but Erkan Zengin touched it on to the post.
Bilic was sent to the stands just before the final whistle, knowing there will be all to play for next week.