Kahveci scored a hat-trick in his side's losing effort, becoming only the third player in Champions League history to score three goals in a game but still have his side suffer defeat, after Ronaldo and Bale did the same.
Brazilian striker Ronaldo scored a treble for Real Madrid in a losing effort against Manchester United in April 2003, while Bale scored three times for Tottenham against Inter in a defeat in October 2010.
After Yussuf Poulsen and Nordi Mukiele gave Leipzig a two-goal advantage to start the game, Kahveci scored just before the half to give the Turkish side life.
Dani Olmo appeared to put the game away for the Germans in the second half, but Kahveci hit back with two more goals – the latter of which was a stunning 85th-minute free-kick – to level the game at 3-3.
All of Kahveci's goals came from outside the box, making the 25-year-old the first player to score a Champions League hat-trick with each goal coming from outside the area since Wayne Rooney's debut in the competition for Manchester United, against Fenerbahce, back in September 2004.
Basaksehir thought they had secured a share of the points but substitute Alexander Sorloth scored a dramatic stoppage-time winner to give Leipzig a 4-3 win and put the pressure on PSG and Manchester United, against who the Germans are competing for a spot in the last 16.
Meanwhile, with the loss, Basaksehir became only the seventh team in Champions League history to score at least three goals in a home match and end up losing.
Leipzig will not have much time to celebrate the victory, as they are back in action on Saturday with a massive Bundesliga clash against champions Bayern Munich. Julian Nagelsmann's men enter the match in second place in the table, two points behind leaders Bayern.
For Basaksehir, they will also return to league action on Saturday when they face Yeni Malatyaspor. Following their league title last season, Basaksehir have begun the current season slowly and are in ninth place.