Zaha is showing no serious symptoms but was not part of Palace's travelling squad to Burnley.
Hodgson told Sky Sports: "He has tested positive unfortunately for coronavirus so is self-isolating, and is awaiting the results of the next test. He is not feeling particularly ill, but unfortunately the test proved [positive], and he is following the procedures."
The Palace talisman, who has been in good form this season after a disappointing campaign last year, was set to make his 200th appearance at Turf Moor on Monday night.
Michy Batshuayi has taken the place of Zaha in the Palace side, with their added dept this season meaning it is less of a blow to Hodgson than it would have been in recent years. Eberechi Eze, the summer signing from QPR, has recovered from muscle tightness to start.
It is the third positive test Hodgson has had to deal of late, with Jordan Ayew returning from international duty last month having contracted the virus while Luka Milivojevic tested positive during the most recent break, though the Palace captain is already suspended for tonight's game.