Leipzig needed to better Porto's result against Monaco to stand any chance of UEFA Champions League progression, but it was not going to plan from the moment Alvaro Negredo gave the visitor an early lead from the penalty spot.
The host peppered the Besiktas goal but was frustrated by the outstanding Tolga Zengin in goal. Captain Willi Orban hit the post while Timo Werner and Jean-Kevin Augustin both had goals chalked off for offside on a frustrating night for the hosts, who also had Stefan Ilsanker sent off.
Naby Keita briefly restored hope after firing a low shot past Zengin with just three minutes remaining before Talisca hit back in the last minute to seal an impressive away victory.
The result, coupled with Porto's 5-2 win over Monaco, means Leipzig heads into the UEFA Europa League while Besiktas finishes top with an impressive tally of 14 points and will head into Monday's draw still unbeaten in the tournament.