After coming from behind to win the first leg 2-1, the Belgian outfit produced another display of determination to fight back from a goal down at the Ataturk Olympic Stadium.
Michel Preud'homme's men looked to be heading out when Ramon levelled the tie at 2-2 in the 48th minute, putting Besiktas ahead on the night and on away goals in stunning fashion.
The Brazilian left-back found the top-right corner with a magnificent effort from 30 yards to give Slaven Bilic's men the advantage in Istanbul.
Yet they were pegged back by a cool Tom De Sutter finish 13 minutes later, the striker racing through before beating Tolga Zengin one-on-one.
And the tie was ended when substitute Boli Bolingoli netted a late double before Besiktas midfielder Olcay Sahan was sent off as Brugge kept their hopes of a repeat of their run to the 1976 final very much alive.
Brugge saw an early chance go to waste when De Sutter broke clear in the fifth minute, only for his near-post effort to be turned wide by Zengin.
Gokhan Tore then drew a low stop from Mathew Ryan at the other end with a shot from the edge of the area, before De Sutter spurned another good opportunity by heading wide from Laurens De Bock's left-wing cross.
Besiktas dominated possession without really threatening the visiting defence, with Demba Ba's powerful off-target strike late in the first half summing up a disappointing opening 45 minutes.
But Besiktas' frustration turned to ecstasy just three minutes after the restart, when Ramon broke the deadlock with a stunning left-footed drive that left Ryan helpless.
Zengin was fortunate not to gift the initiative back to Brugge when his clearance bounced off De Sutter and wide of the goal moments later.
However, after Mustafa Pektemek had blazed over at the other end, De Sutter finally got his goal as he latched on to Timmy Simons' header forward and confidently slotted home.
Besiktas pressed to try to force the tie into extra time but with the play stretched Bolingoli took advantage by rounding Zengin and rolling the ball into the empty net 10 minutes from time.
Bolingoli dispossessed Necip Uysal and stroked beyond Zengin in the 90th minute to add further gloss to an excellent away performance, with Sahan then dismissed in added time for lashing out at Obbi Oulare.
