Arsenal midfielder Ramsey was given his marching orders 10 minutes from time in Istanbul on Tuesday for the first time in his club career for a second bookable offence after pulling Oguzhan Ozyakup's shirt and will miss the return clash at the Emirates Stadium next Wednesday.
Both sides had their chances in a scrappy encounter and substitute Alex Oxlade-Chamberlain came closest three minutes from time when his strike came back off the post, but defeat would have been harsh on Besiktas.
Demba Ba almost gave the Turkish side, who were backed by a hostile crowd at their temporary Ataturk Olympic Stadium home while the Vodafone Arena is being redeveloped, a stunning lead with a speculative lob right from the kick-off and was a real handful for the Arsenal defence.
Besiktas will have to spring a major surprise in order to progress, as Arsenal have reached the group stage 16 years in a row and have won all 12 Champions League qualifying matches in which they have featured in.
Nacho Monreal and Olivier Giroud replaced hamstring injury victims Kieran Gibbs and Yaya Sanogo in the Arsenal starting line-up, while Besiktas midfielder Atiba Hutchinson was also ruled out due to a hamstring problem.
Ba was clearly brimming with confidence after his hat-trick in the second leg of the third qualifying round tie against Feyenoord and the striker came close to a stunning opening goal straight from the kick-off.
Wojciech Szczesny was almost caught out by an audacious lob from the Senegal international from just inside the Arsenal half, but tipped the ball against the crossbar after frantically back-pedalling.
Ba was in the thick of the action and forced another save from Szczesny eight minutes in with a left-footed volley.
Arsenal started to get into their stride and Giroud had an effort blocked on the turn after good work from Alexis Sanchez before Santi Cazorla fired wide of the far post.
Ba was holding his head two minutes before the break after Calum Chambers slipped and the striker cut into the penalty area, but fired just wide of the near post. Giroud then missed his kick at the other end in a dramatic end to the first half.
Ramsey linked up well with Giroud just after half-time, but the Wales midfielder's touch let him down and the chance had gone.
And he should have been made to pay moments later, when Olcay Sahan raced away and cut inside before guiding a right-footed effort narrowly wide of the far post.
Arsenal then lost captain Mikel Arteta to injury following a challenge from Ba, with Mathieu Flamini replacing the Spanish midfielder five minutes into the second half.
Neither side played with any fluency in a scrappy second half, but Mathieu Debuchy caused concern in the Besiktas defence when he cut the ball back across goal following a marauding run, but the Turkish side cleared.
Cazorla tested Tolga Zengin with a late free-kick that came through the wall, then Ramsey saw red for a tug on Ozyakup, but Besiktas were unable to make their numerical advantage count.
And Oxlade-Chamberlain almost snatched a winner two minutes from time, when he cut inside and saw his left-footed effort tipped onto the post by Zengin.
