Bereft of the talismanic Ronaldo, who is self-isolating after testing positive for COVID-19 while on international duty with Portugal, Juve looked far from its best at Stadio Ezio Scida.
Alvaro Morata's first goal since rejoining the club on loan from Atletico Madrid cancelled out Simy's early penalty for the hosts, but their hopes of going on to claim all three points were dealt a blow with 30 minutes remaining when Chiesa, who joined on loan from Fiorentina on transfer deadline day, received a straight red card.
Juve, whose veteran goalkeeper Gianluigi Buffon made his 650th Serie A appearance, thought it had won it late on thanks to a close-range finish from Morata but he was deemed to be offside after a lengthy VAR (video assistant referee) review.
Crotone started with a spring in its step and was rewarded after 11 minutes when Simy stroked home from the penalty spot after Leonardo Bonucci had been adjudged to have fouled Arkadiusz Reca inside the area.
Juve bounced back nine minutes later when Morata slid home from four yards after being picked out by Chiesa.
Manolo Portanova – making his first appearance since May 2019 – was denied by Alex Cordaz at his near post shortly before the interval, while at the other end Buffon denied Pedro Pereira.
Rodrigo Bentancur flashed an effort wide immediately after the restart before Luca Cigarini whipped wide of Buffon's left-hand post from a promising position on the edge of the area.
Chiesa then saw red for a late challenge on Cigarini, although Andrea Pirlo's men almost stole ahead soon after when Morata flicked a header onto the post.
The Spain striker was denied a second inside the closing 15 minutes, with a VAR review showing that he was marginally offside when he turned home Juan Cuadrado's low ball into the area.
A point moves Juve up to fourth, four points behind leader AC Milan – which claimed the derby over Inter Milan 2-1 earlier in the day – while Crotone is off the bottom after picking up its first point of the season.