The visitors trailed to Alberto Paloschi's 15th-minute goal, but would have levelled 10 minutes later had Amauri not seen his penalty saved by Christian Puggioni.
They had to wait just three more minutes for the equaliser, though, as Cassano finished with aplomb.
And the Italy striker turned creator to set up a 93rd-minute winner for Alessandro Lucarelli, which left Chievo in trouble towards the bottom of the table.
Marco Parolo shot just wide for Parma early on, but they had a let off in the 13th minute when Alberto Paloschi's goal was ruled out, Cyril Thereau, who set him up, adjudged offside.
But the hosts went ahead soon after following a scramble in the Parma area.
Simone Bentivoglio struck the bar from close range and on the rebound Paloschi tapped in.
Amauri wasted his chance from the spot after Perparim Hetemaj had been penalised for handball, but Cassano brought Parma level.
The striker fired into the bottom left corner from inside the area.
Amauri, seeking to atone for his penalty miss, headed narrowly wide and then almost caught out Puggioni with a clever lob, which the goalkeeper tipped just over, as Parma ended the half in charge.
And so it continued in the second, Amauri, Cassano and Dejan Lazarevic all shooting over the bar.
Amauri was heavily involved now and he fired over the top again after a run down the left channel before firing another effort wide.
Gennaro Sardo almost snatched an undeserved winner in the 85th minute for Chievo, but his shot from long range went inches wide.
Instead Parma stole the points at the death, Cassano's trickery down the left setting up Lucarelli, who shot home.