Stoke had been leading through Ricardo Fuller’s first goal in European competition, but Besiktas had in truth dominated before their second-half comeback.
A man to the good, Besiktas went on to win in comfort and style as substitutes Mustafa Pektemek and Edu added two more, the latter a fine strike.
The result means the Potters now run the risk of facing one of the higher-ranked Champions League drop-outs in the last 32, while Besiktas’ elevation to the top of the group means they will be amongst the seeded teams.
With Stoke already qualified, Tony Pulis left behind the entire XI that beat Tottenham on Sunday, naming only three midfielders and a reserve keeper on the bench.
Besiktas, meanwhile, only needed to avoid defeat to ensure qualification ahead of Dynamo Kiev, who themselves needed victory at home to Maccabi Tel Aviv and a five-goal swing.
It seemed the Turks would at least get their point as they bombarded a reluctant Stoke side from the off as Fabian Ernst, Veli Kavlak and Filip Holosko all forced fine saves from Asmir Begovic in the Potters’ goal.
But they were not counting on the hunger of Cameron Jerome and Ricardo Fuller, both with a point to prove to Pulis as they battled endlessly from the front.
Jerome had fired a warning shot when Kenywne Jones sent him clean through, with veteran goalkeeper Rustu Recber coming to the host's rescue, and just before the half hour they took the lead when Fuller’s shot took a wicked deflection off Egemen Korkmaz to fly past the stranded former Turkey stopper.
The deflection completely bamboozled the ex-Barcelona man but Fuller was happy to claim his first goal in Europe and his 50th for Stoke, on his first start this season after injury. Jones, meanwhile, got the assist in a Caribbean combination.
The Jamaican’s determination was shown once more when he closed down Rustu following a poor Korkmaz backpass, but his block hit the post and the covering defender put the rebound wide - and Fuller into the post.
That caused problems, with Higginbotham seemingly having forgotten how to play wide, but Jerome covered for him on a couple occasions, while Begovic again saved well as Fernandes had a pop from range.
Dynamo, meanwhile, were doing their bit to keep the group interesting - with Besiktas trailing at the break, they were 2-0 up at home to the Israelis.
The second half started in nondescript fashion in Istanbul, but in Kiev there was a boost for Besiktas as Maccabi pulled one back, meaning a further three-goal swing was needed.
With the game becoming plain sailing for the Potters - who at this stage were comfortably winning the group - Upson showed why he is no longer first choice for club or country getting caught far too high up the pitch defending against Almeida.
Almeida, himself no Usain Bolt, comfortably beat him for pace, with Upson launching into a tackle from behind that earned Besiktas a penalty, and the Stoke defender a straight red card.
Fernandes dispatched the spot-kick with aplomb, sending Begovic the wrong way, and calming the Turkish side’s nerves, with a point sufficient send them through, whatever Dynamo managed to conjure up.
With Stoke a man down the Turks sensed group victory and, after a concerted spell of pressure, substitute Pektemek headed a Fernandes corner past Begovic for the lead on 74 minutes.
The visitors were then fortunate not to go down to nine men when Fuller - booked in the first half - escaped a second yellow when referee Marcin Bolski gave him the benefit of the doubt after a blatant dive in the box.
But they suffered further when Edu, on for Almeida, picked up the ball on the right-hand side of the box and fired an unstoppable, rasping effort across Begovic and into the top left.
It was a wonderful goal to add a gloss to a fine run of form for Besiktas, who have now won five of their last six matches in all competitions.