Bruno Fernandes scored United's first after seven minutes with a rasping half volley from 20 yards out.
The Portugal international had his second shortly after when Mert Gunok dropped a routine cross at the midfielder's feet.
Marcus Rashford was brought down in the box and then converted from the penalty spot to put United ahead 3-0 after 35 minutes.
It was the first time United had lead by three goals at half-time since 2007.
Basaksehir did get a goal back through Deniz Turuc's curling free-kick.
But Daniel James came off the bench to score his first goal for United since March to round off the victory.
United are now top of Group H on nine points, three ahead of PSG and RB Leipzig.
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