The 33-year-old Argentina international chose to play through the pain after a physio examined his finger, despite the risk of aggravating the injury ahead of his country's World Cup title defence starting next month.
He made two saves as Villa cruised to victory, securing their first major European trophy since 1982 and first major silverware since their League Cup win in 1996.
"Today I broke my finger during warm-ups, but I didn't see it as a bad thing," Martinez told ESPN after the match, laughing off the injury.
"I've never broken a finger before and every time I tried to catch the ball, the finger would just slip away in the other direction. But, these are just things you have to go through."
Martinez, whose penalty shootout heroics helped Argentina win the World Cup in Qatar in 2022, has been at Villa since 2020, making over 250 appearances for the club.
"Every time I step out to defend the Aston Villa goal, I do so with immense pride," he said.
"Today, I tried to pour in all my experience, and all those training sessions that no one ever sees and finally bore fruit today."