Ryan Babel netted an early opener before Anderson Talisca put the hosts in full control prior to half-time to end a run of six matches without a victory in the competition on their own turf.
Vodafone Park was draped in darkness when the floodlights went out midway through the second half, the halt in play only serving to stifle any chances of a late Leipzig response.
In becoming the first German team to lose a Champions League match on Turkish soil, last season's Bundesliga runners-up suffered further misery with star forward Timo Werner subbed off after just half an hour.
Senol Gunes' side, however, appear well on course to reach the knockout stages for the first time in their history as they boast two wins from two in Group G.
Besiktas set a frenetic early tempo and almost went ahead in just the fourth minute as Talisca challenged Peter Gulacsi from Quaresma's right-sided delivery, the ball dropping just wide of the near post.
Gunes' side only had to wait a further seven minutes to go 1-0 up, though, as Babel clinically swept Cenk Tosun's clever cut-back into the bottom-left corner.
Marcel Sabitzer attempted to restore parity by giving Besiktas goalkeeper Fabri his first piece of work with a long-range effort, but the visitors' task grew tougher when the in-form Werner was withdrawn just after the half-hour with an apparent injury.
Gulacsi was next forced to beat away Tosun's strike from a narrow angle as the home side looked to extend their advantage before the break, and they did just that as Talisca met Quaresma's outrageous outside-of-the-boot cross with a thumping header.
Sabitzer remained the biggest threat for Leipzig after the restart and twice drew superb saves from Fabri, the Spanish shot-stopper palming the first away from the bottom-right corner before clawing another goal-bound effort around the other post just moments later.
Substitute forward Bruma then flashed a firm shot wide as Ralph Hasenhuttl's men set about reducing the deficit, only for their momentum to be stalled when the floodlights went out.
Leipzig's momentum stuttered as the players remained off the pitch for around 10 minutes, the German outfit restricted to half-chances on the resumption.
Austria international Sabitzer did have one last chance to finally get on the scoresheet when he fired against the crossbar, while Jean-Kevin Augustin blazed over with Leipzig's final chance of an entertaining contest.