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.
They 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.
Two over the weekend, two today. Moise Kean gets his second 💫 pic.twitter.com/nxKIilSWgp— Champions League on CBS Sports (@UCLonCBSSports) October 28, 2020
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.
What does it mean? PSG, like Neymar, look limp
PSG have their first three points in Group H but this display will not have convinced many onlookers that they are capable of matching last season and reaching the final again.
The loss of Neymar for any period of time would also be a significant blow as their hectic schedule continues towards the winter months in France.
Kean shows clinical touch
He might have had just 20 touches and seven successful passes, but Kean provided the killer touch to break Basaksehir's resistance with two well-taken finishes in the second half.
Epureanu deserved better
Alexandru Epureanu was defensively superb, making six clearances, three interceptions and three blocks, but he could only watch as Kean glanced home the opening goal after letting him out of his sight.
PSG face Nantes in Ligue 1 on Saturday before next Wednesday's visit to RB Leipzig. Basaksehir meet Konyaspor on Sunday, then head to Old Trafford to meet Manchester United.