The Italy forward scored a peach of an opening goal in the second half but saw his effort cancelled out by Alvaro Morata's equaliser.
Neither side could find a winner in extra time after an enthralling game and an exhausted Chiesa reflected on the win.
Leonardo Bonucci said the win over Spain for Italy was the toughest game of his career
The Azzurri came out on top in the shootout to book their place in the semi final
He told beIN Sport: 'I cannot describe my emotion by words, it was a tough match, Spain were great tonight but we come back here on 11 July for the final. We will see, we will see.
'Spain were great, they have star players