Cenk Tosun scored the decisive goal in the 59th minute, hot on the heels of having smashed a shot against the Tottenham crossbar at Istanbul's Ataturk Olmypic Stadium, where floodlight failures caused lengthy delays at either end of the match.
Mauricio Pochettino's visitors had been the better side for much of the contest up until the winning goal, although much-maligned striker Roberto Soldado passed up chances on each side of the lights going out for the first time.
A draw would have been sufficient for Tottenham to emerge as group winners, but the hosts enjoyed the better opportunities after going ahead, with Kerim Frei wasting the pick of them before a second plunge into darkness early in stoppage time halted any potential late rally in its tracks.
The first opening fell Tottenham's way as Nacer Chadli's cross on the end of Moussa Dembele's fine throughball narrowly evaded Soldado.
The first unscheduled interruption arrived a minute later, but the visitors remained the most fluent team following a 16-minute delay.
Soldado scuffed a glorious close-range chance in the 16th minute when Younes Kaboul knocked down a Paulinho free-kick and the Spaniard had a shot blocked by Ersan Gulum after the impressive Dembele picked him out.
In the 25th minute Chadli chipped home, but was denied by the offside flag, before Besiktas winger Olcay Sahan bustled forward and fired a shot over at the second attempt after Kyle Walker - playing a Tottenham first-team game for the first time in nine months following an abdominal problem - blocked his initial effort.
A quiet opening to the second period spung to life in the 58th minute when Tosun pounced on an error from Kaboul to chest down and thrash a venomous 25-yard strike against the crossbar.
That chance unsettled Tottenham and Tosun was celebrating a few moments later as Sahan touched Gokhan Tore's right-wing cross back across goal to leave the forward with a simple finish.
Tosun marked the goal by displaying the shirt of Mustafa Pektemek after his team-mate was hospitalised with a facial injury sustained during the weekend win over Trabzonspor.
Soldado could not convert from a corner as Tottenham sought an instant response and Danny Rose's low drive was saved by home goalkeeper Tolga Zengin.
Besiktas could have given themselves breathing space in the 67th minute when Sahan's flick sent Frei racing clear of the away defence, but the former Fulham midfielder fired into the side-netting.
Slaven Bilic's team continued to threaten further misery for Tottenham as Jose Sosa cut inside Vlad Chiriches to shoot over.
By the time the lights failed again, Tottenham's prospects were already looking dim and Pochettino must reflect on three matches without a goal in all competitions.
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