Missed chances have plagued Newcastle's recent form and again they were wasteful in front of goal in the first half against Leicester, mustering 12 shots without finding the net, with Sean Longstaff the guilty party on two occasions.
The hosts almost edged in front immediately after the restart, but Joelinton's prodded effort came out off the post, via a Marc Albrighton intervention.
It was left to Burn to finally break Leicester down, with the Newcastle-born fullback finishing well on the hour mark to raise the roof at St James' Park.
The home supporters did not have to wait long for a game-clinching second, as Joelinton made one count to make sure of the win and keep Newcastle in the hunt for their first major domestic trophy since 1955.
"It would mean everything to win a trophy but we just need to focus on the next game, take it one game at a time and now we look to the Premier League," defender Kieran Trippier told Sky Sports.
"We knew this was important because we went out of the FA Cup, the lads were brilliant tonight."
Leicester arrested a run of three successive defeats with a victory over fourth-tier Gillingham in the FA Cup at the weekend, but they never looked like making it successive wins.
"The best team won," coach Brendan Rodgers told the BBC. "We are disappointed with the goals we conceded. We gave away two soft goals. We lacked quality in our play if I'm honest but the players gave everything.
"We had a few chances in the game, good chances we could have scored from but if you don't take those opportunities it is tough. I am focussed on the players that are fit and available. That is where we are at."