The ball bounced four times on the rim before dropping in as the Raptors won 92-90 to claim the series 4-3, BBC Sport reported.
They will now play the Milwaukee Bucks in the Eastern Conference finals.
"It was short, then it was long, then it was in," said Toronto's Marc Gasol of Leonard's winner.
Teammate Fred VanVleet added, "It was like a movie moment."
The Raptors are attempting to reach the NBA Finals for the first time in their history.
They signed Leonard from the San Antonio Spurs, with whom he won the NBA in 2014, in July 2018.
"I'm a guy that acts like I've been there before. So probably the last time you've seen me scream was when we won [in 2014]," said Leonard, who finished with 41 points.
"Whenever it's a moment that I haven't really experienced, I'll try to show some emotion and let it come out. Tonight was one of those nights. It was a great feeling."
Joel Embiid, who was in tears after the 76ers lost, added, "I felt like we had a chance. A lot of things go through your mind and it just sucks."
Elsewhere on Sunday, the Portland Trail Blazers beat the Denver Nuggets 100-96 to win the series 4-3 and set up a Western Conference meeting with reigning champions the Golden State Warriors.
In that match the Warriors' Steph Curry and the Trail Blazers' Seth Curry will become the first brothers to face each other in the NBA conference finals.
Their mother Sonya said she will ‘flip a coin’ to decide who to support in each match.
She told ESPN, "It's going to be so much fun. From our end, to see both of our sons compete at this level and for the goal to be a championship is such a blessing. We never could imagine this.
"One of them might go home. But we're going to the championship."