In a pulsating fixture in Naples, the visitor had twice squandered the lead with Adriano Correia and Aboubakar's first cancelled out by Dries Mertens and a Manolo Gabbiadini penalty respectively.
However, there was late drama when Aboubakar nodded in from close range to earn a crucial away victory for the Turkish champion in a tight Group B.
Napoli will feel aggrieved at letting the match slip having enjoyed the better of the game on home soil, and Lorenzo Insigne's missed penalty early in the second half with the score at 2-1 proved costly.
For Aboubakar, who pounced on a Jorginho's mistake to make it 2-1, it marks a return to form for a striker that has struggled since moving to Besiktas on loan from Porto in August and he had failed to score prior to the encounter.
His first goals have helped lift Besiktas to second in the pool, though, just one point adrift of Maurizio Sarri's side - which has six points from three matches.
Napoli made a bright start and a clever corner routine involving Faouzi Ghoulam and Insigne led to a chance for Mertens in the third minute, but he blasted over the top.
Besiktas went in front with its first meaningful attack, though. Tolgay Arslan set Ricardo Quaresma free down the right and the Portugal international's low cross to the back post found Adriano, who beat Manuel Reina from close range.
Unperturbed, Napoli continued to dominate and deservedly equalised on the half-hour mark. Jose Callejon sprung the offside trap to meet Marek Hamsik's ball over the top and he drilled in a low cross for Mertens to tap home inside the six-yard box.
However, Besiktas caught Napoli cold again seven minutes before the break. Jorginho's poor pass was easily picked off by Aboubakar and he slotted his shot past Reina.
Napoli had a great chance to level early after the break when referee Sergei Karasev pointed to the spot after Fabricio brought down Mertens inside the area, however the goalkeeper made amends for his error by pushing Insigne's tame effort wide.
However, the host had another chance from the spot in the 69th minute when Caner Erkin was adjudged to have hauled down Mertens inside the box and Gabbiadini fired home the resulting penalty.
Gabbiadini had the ball in the net again shortly after, but the play had already been brought back for offside.
And Besiktas made the most of the reprieve four minutes from time as Aboubakar headed past a helpless Reina from Quaresma's free-kick.