Leipzig, last season's surprise semi-finalists who are unbeaten in all competitions so far this season, went in search of a quick goal and after a few early chances Angelino put them in the driving seat in the 16th minute.
The 23-year-old, on loan from Manchester City since January, turned into an accomplished centre forward when he beautifully controlled a Kevin Kampl cross into the box with a fine first touch, pivoted on the spot and fired in for the lead.
He completed a quick counter attack four minutes later after the Turks lost possession in their own half.
Angelino, who has already scored twice in four Bundesliga games, came close to a hat-trick a little later but his powerful volley straight from a corner kick whizzed past the post.
Basaksehir, making their maiden group-stage appearance, managed to keep up for the opening 20 minutes but were then on the back foot.
They controlled possession in the second half but failed to carve out any real scoring chances against the disciplined German backline.
Leipzig have three points in Group H, along with Manchester United who beat Paris St Germain 2-1. The Germans next play United in Manchester on Oct. 28.