Three players were sent off in the second-half including Zaza who had opened the scoring after 52 minutes after connecting with a perfect low cross from Armando Izzo for his third goal this season.
But the Italian forward was sent off on 74 minutes for arguing with referee Massimiliano Irrati over a free kick awarded against him.
Cagliari pulled level a minute later when Leonardo Pavoletti headed in a Luca Cigarini cross, with the goal confirmed following a long VAR review.
But there was more late drama with Cagliari having two players sent off for second bookings.
First Luca Pellegrini went off with eight minutes to go followed by Nicolo Barella deep into injury time.
Cagliari goalkeeper Alessio Cragno did well to deny Daniele Baselli from sealing a late win for Torino.
Torino stay seventh, five points off AC Milan who occupy the final Champions League spot, with Cagliari moving up to 11th.
Juventus missed the chance to seal an eighth consecutive league title with six games to spare after a 2-1 loss at SPAL on Saturday, just their second defeat this season.
The champions have a 20-point advantage on second-placed Napoli and can still seal the title later Sunday if Carlo Ancelotti's side fail to win at bottom club Chievo.