Leonardo Bonucci's mistimed tackle saw Simy convert a penalty, cancelled out by Morata meeting a Chiesa assist.
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Despite having a player sent off Juventus got a goal in the 76th minute only for Morata's effort to be ruled out for offside.
A point moves Juve up to fourth, four points behind leaders Milan, while Crotone are off the bottom after picking up their first point of the season.
Just as they did in the 2-2 draw at AS Roma three weeks ago, Juve had an uphill battle going down a man after Chiesa's exit.
"Unfortunately, we went down to 10 men in the last two games because of our errors, but we are a young team that needs to work and there hasn't been much time for that with so many internationals", he told DAZN.
They will be without talisman Cristiano Ronaldo on Saturday, after the Portuguese forward tested positive for coronavirus.
"We made a few mistakes in the timing of the press, but as I said, these things happen when we're still getting the machine oiled".
"We knew there would be many teams fighting for the title, it will be a battle day-by-day", said former Juve midfielder Pirlo. "Unfortunately we are losing some on the road but we will be back in front soon".