A positive result for Thomas Tuchel's side, who lost 2-1 to Manchester United on matchday one, was undermined by the loss of Neymar to an apparent thigh injury in the first half.
PSG delivered a largely laboured display in Turkey, one that is unlikely to do much to ease some of the pressure on Tuchel, but Kean followed up his double against Dijon last weekend with a timely header from Kylian Mbappe's corner 64 minutes in and a clever finish 11 minutes from time.
Neymar hobbled off after treatment before the half-hour mark and his absence was telling as PSG mustered only one shot on target in the first 45 minutes.
PSG had Keylor Navas to thank for two saves to deny Edin Visca as Basaksehir grew in confidence going forward while remaining resolute at the back.
Kean's breakthrough came in disappointing fashion for the hosts, though, the striker simply nodding Mbappe's cross past the static Fehmi Mert Gunok after being left unmarked.
Navas made a smart save from Deniz Turuc's half-volley and PSG almost doubled their lead on the break, Mbappe eventually turning and shooting over the crossbar after starting a move in his own half.
Kean made the result safe with 79 minutes gone, volleying on the turn beyond Gunok from six yards out after Mbappe's good work in the penalty area.