Andrea Stramaccioni's men took their time to get going against the Serie B outfit, but Antonio Cassano and Fredy Guarin both scored after the break to guarantee a last-eight meeting with Napoli or Bologna, who play on Wednesday night.
But with 10 minutes of normal time remaining - and with all three of Stramaccioni's substitutes in play - Inter goalkeeper Luca Castellazzi fell awkwardly on his shoulder, prompting ex-Genoa playmaker Palacio to pull on the gloves for the first time since his childhood.
The Argentinian told Inter's official website: "I made a great save from (Alessandro) Carrozza! We were leading 2-0 so we felt a bit comfortable but it was important to win the match.
"Why did I go in goal? Initially (Cristian) Chivu said that he'd do it, then Stramaccioni decided he wanted either me or (Ricky) Alvarez.
"Luckily it was only 15 minutes! Had I done it before? Yes, a few times as a kid.
"On Saturday it's Genoa, it'll be a special game for me, but now I'm an Inter player. It won't be an easy match.
"What do I want from 2013? To win the league."
Stramaccioni was content with the manner of victory over Andrea Mandorlini's promotion-chasing side.
He said: "Verona made a great start to the match, I think they're one of the best sides in Serie B and have a great coach.
"We changed the tempo at the beginning of the second half and opened up a two-goal lead.
"Palacio in goal? With his stature he made the backline feel safe."
Stramaccioni then gave an update on Castellazzi's injury, adding: "He'd picked a knock when he made his first save, but the medical staff told me not to worry."