Motta denied Gianluca Scamacca, Davide Zappacosta, Mario Pašalić and Charles de Ketelaere from the spot as Lazio won 2-1 in the shootout after the tie finished 3-3 on aggregate.
The young goalkeeper, who only joined the club from Serie B side Reggiana in January, was seen in tears at full time.
Speaking to TG1, he was candid about the emotion: "I let myself go a bit with emotion, something I never want to show. Staying strong defines the person you are more than being overly emotional."
His path into goalkeeping began young. "I liked being a goalkeeper and throwing myself into the gravel. My father thought I was a bit crazy, but I didn't feel any pain."
His idol growing up was former Chelsea and Czech Republic goalkeeper Petr Čech. "I told my parents I wanted to be like him, that I wanted to try. They gave me strength and made so many sacrifices to make sure I'm here today."
On his ambitions going forward: "I never want to stop learning, improving, and growing. I want to try to reach as high as I can, without regrets or remorse, enjoying life without taking anything away from myself."
Lazio face Inter Milan in the Coppa Italia final on 13 May.