The 24-year-old American was presented with the trophy after Chicago rallied in the final minutes to beat the Boston Bruins 3-2 and clinch the best-of-seven series 4-2.
Kane was Chicago's leading point scorer during the play-offs with 19 points from 23 games, including nine goals, but said he felt lucky to win the award, insisting his teammates did all the real work.
"We actually came up with a name for myself this morning, calling me the 'Benefish,' for the beneficiary of all their hard work," he said.
"I had a couple chances to finish and ended up doing that, so got to give them the credit."
Despite his reluctance to take credit for his achievements, Kane's star has been rising for years.
A native of Buffalo, New York, he was drafted first overall in 2007 after an outstanding junior career and was named the NHL's rookie of the year in 2008.
He won a silver medal with the American team at the 2010 Vancouver Winter Olympics then his first Stanley Cup with Chicago later that same year, scoring the championship-clinching goal in overtime against the Philadelphia Flyers.
"I think there's something about our core. Hopefully we can stay together a long time because that's two Cups in four years, and we seem to only be getting better and better," he said. "It's unbelievable to be in this situation.
"It's just a great group and it's going to be fun to celebrate with them."
Kane paid a special tribute to Chicago captain Jonathan Toews, who won the Conn Smythe trophy in 2010, and played in Monday's Game Six despite saying he had his "bell rung" in the previous game.
"He's a great player. He's played big in a lot of big games," Kane said. "That's really all you can say about Jonathan Toews is he's a competitor. He leads the team in the right way and we all follow."