The Belgium midfielder rifled an unstoppable effort into the top corner from 25 yards, as Brendan Rodgers' Foxes put another genuine stamp on years of recent quality.
Former Leicester full-back Ben Chilwell saw a last-gasp Chelsea effort ruled out for offside after a video assistant referee review, while Kasper Schmeichel earlier made a superb save to deny Mason Mount.
Tielemans, who was awarded man of the match, told reporters after the game that his stunning goal was simply what he was asked for before the game.
"It feels really good, I think it is amazing," Tielemans told the BBC.
"I am really glad to have scored the goal to have helped the team to win the game.
"It was a really tight game, but to score a goal like that in the final, what a goal it was.
"Today I got a text message saying I had to hit for the top corner - and I did it."
Tielemans added: "To have the fans in today was really special, I hope everyone will savour today and also the fans who have been at home and supported us the whole year through."
Additional reporting by PA.