Basaksehir lost at home but fought well, scoring 3 goals.
Istanbul club's Turkish midfielder Irfan Can Kahveci was on fire as the 25-year-old netted three goals to keep Basaksehir's hopes alive.
RB Leipzig earned the critical win as Danish forward Yussuf Poulsen, French defender Nordi Mukiele, Spanish forward Dani Olmo and Norwegian forward Alexander Sorloth scored for the 4-3 win.
The match was 3-3 until Leipzig's very late winner.
Sorloth has scored the winning goal for RB Leipzig in the 92nd minute as he fired outside the area.
His goal ended Basaksehir's Round of 16 hopes.
In the Group H, Basaksehir are in the last position with 3 points in 5 matches.
Third-place Paris Saint-Germain (PSG) have 7 points and the French team are currently playing against the Group H leaders, English club Manchester United, who bag 10 points.
Leipzig had 9 points to come second.
The Turkish team could not enter the second-tier UEFA Europa League knockout stage if PSG had a point on Wednesday.
Basaksehir will visit PSG next week as the group stage will be over.
Results
Group E: Krasnodar - Rennes: 1-0
Group H: Medipol Basaksehir - RB Leipzig: 3-4
