Leonard scored 22 points and grabbed 10 rebounds in San Antonio's 104-87 Game Five win in the best-of-seven series as the 22-year-old became the second youngest MVP in the NBA Finals history.
Magic Johnson picked up the honour as a 20-year-old rookie for the Los Angeles Lakers in 1980, and again two years later, while Spurs team mate Tim Duncan won it as a 23-year-old in 1999.
After initially struggling to make a mark in the opening contests of the series, Leonard stepped up his defense against back-to-back MVP LeBron James and scored over 20 points in the next three games.
An elated Leonard was mobbed by team mates after commissioner Adam Silver announced his name as the winner of the Bill Russell MVP Award, with the player admitting he was shocked to be given the prestigious honour.
"At the moment, I was just happy," Leonard told reporters. "Just had faith throughout the whole game, but I didn't think at all I was about to win the MVP of the Finals."
Leonard averaged 17.8 points, six rebounds and 1.6 steals per game in the 4-1 series victory over the Heat.
"The first two games he didn't play so well," James said. "I thought he attacked more in the last three games, shot the ball extremely well."
San Antonio coach Gregg Popovich felt the award was a deserved culmination of the hard work Leonard had put in to improve his game.
"He walks the walk. I mean, he is there early; he's there late. He wants more. He wants me and the coaches to push him," added.