Burn's second-half strike, plus a calm finish from Joelinton sealed the Magpies' place in the last four.
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Eddie Howe's side put sustained pressure on the Foxes, and their efforts soon paid off on the hour mark.
Centre-back Burn, who has supported Newcastle since a young age, was able to skip past two Leicester players as he broke into the box.
He then hit a low finish past Danny Ward and into the bottom right corner to give his side a deserved lead.
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Amid the celebrations from the home crowd, talkSPORT's Adrian Clarke said: "This is a majestic goal from Dan Burn, what a way to open your account.
"What a reception he is receiving, he has been outstanding in the game. Leicester were messing around with the ball inside their own defensive third.
"He just bursts past two players, three players, it opens up on his right foot. Has he got the confidence to slip it into that far corner with his weaker foot? Yes he has!"
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Burn is in his second season at St. James' Park having signed from Brighton last January.
He had failed to score in his previous 36 appearances for the club but is now off and running for his local side.
Just 12 minutes later, Joelinton made it 2-0 to Newcastle with a cool first-time finish, his third of the campaign.
They are now on their way to the semi-finals of the Carabao Cup, being one of four Premier League sides fighting for a spot in the final.
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