Chiesa, signed from Fiorentina during the recent transfer window, was given a straight red card for a studs-up challenge on Luca Cigarini on the hour.
Juventus, missing Cristiano Ronaldo after he tested positive for COVID-19 on Tuesday, thought they had scored a winner when Alvaro Morata turned the ball in from close range in the 77th minute but a long VAR review showed his heel was offside.
Crotone had taken the lead with a Simy penalty in the 12th minute before Morata levelled from Chiesa's assist nine minutes later. Juve, chasing a 10th straight title, have eight points from four games while Crotone picked up their first point. (Writing by Brian Homewood; Editing by Ken Ferris)