Boyhood Magpies fan Burn's unforgettable first for the club since coming home to join the club a year ago lifted the roof off St. James' Park with 60 minutes gone, before Joelinton's excellent finish 12 minutes later put the game beyond the Foxes and made sure of Eddie Howe's charges' place in the last four.
United were irrepressible throughout against Brendan Rodgers' side, and will now face a two-legged semi-final - their first since 2005 - with the prospect of a first Wembley appearance since 1999 now tantalisingly close.
It took just 40 seconds for the hosts to demonstrate their intent. Bruno Guimarães clipped a pass to Miguel Almirón in the Newcastle half, with the Paraguayan breaking at speed. After a 40-yard sprint he played in Joe Willock, whose cross was perfect for Sean Longstaff, but the midfielder bounced his volley over the bar.
It was all Newcastle early on, with the Foxes pinned back. Guimarães blasted a loose ball wide and the returning Callum Wilson saw his effort from Almirón's cutback deflected past the upright, while at the other end, former Magpie Ayoze Perez fired straight at Nick Pope in a rare Leicester attack.
Chances kept coming and, to Howe's men's frustration, going. Guimarães drilled another wide from just outside the area, almost exactly where he'd had his first shot from minutes earlier, and Longstaff forced a terrific save from Foxes goalkeeper Danny Ward, who reacted well to palm a goalbound shot from 10 yards out away for a corner.
Patson Daka then bizarrely opted to square the ball instead of shoot when he got in behind the otherwise steady United backline, but there was never really any danger of the visitors breaking the deadlock. Before the end of the half, Almirón and Guimarães both forced Ward into saves and just 20 seconds after the restart, Joelinton cut inside from the left and struck the base of the post with a shot that took a nick off a Leicester defender on its way to goal.
Tielemans' wayward half-volley briefly worried Newcastle but the hosts still dominated. Burn nodded a Kieran Trippier free kick over and Longstaff was denied by another excellent Ward stop, but a goal was coming - and on the hour, they got it. It was beautifully taken by lifelong United supporter Burn, who burst into the box from the left before finishing low with his unfavoured right foot, sending the Gallowgate End and all those in black and white wild.
Brendan Rodgers responded by sending on Jamie Vardy and Kelechi Iheanacho to supplement the Foxes' forward line, but it had little impact. Ten minutes after Burn's memorable strike, Joelinton doubled the lead, capping a fine team move which started with a throw-in in Newcastle's own half. Trippier found Almirón in space, and his through ball for the Brazilian was perfect. Joelinton took it first time, firing across Ward into the same corner Burn had found earlier to give United what looked to be an unassailable advantage.
Just minutes later, though, they were issued a warning as the visitors carved out two golden opportunities. Both fell to substitute Vardy and the ex-England striker wasted both, dragging the first one wide when one-on-one with Pope before bobbling one past the upright after being sent clean through on goal.
But despite those short-lived scares, it was a comfortable night for Howe's men. Willock went close towards the end, curling wide, but the scorelined remained the same. This was a richly deserved win for Newcastle, who are now potentially just two games away from a long-awaited trip to Wembley.
