Neil Winskill's youngsters raced into a 3-1 lead before the half-time break courtesy of Dylan Stephenson's hat-trick, with the striker breaking the deadlock in the opening stages with a cool close-range finish.
Three goals were then netted in the dying minutes of the first-half, with Stephenson doubling United's lead at Whitley Park with a fierce low drive before Leeds responded in quick fashion through Joseph Snowdon's finish.
There was enough time, though, for Stephenson to net his and United's third before the half-time whistle following his bundled effort on the stroke of half-time.
Joe White extended the Toon teenagers' advantage in the second-half to seal an emphatic win as Neil Winskill's side, watched on by United head coach Steve Bruce, set up a home clash against Watford in round five.
Newcastle had already beaten Yorkshire opposition to seal their place in round four round after a third round victory at Huddersfield Town and they got off to a dream start against their Whites counterparts in the 13th minute through Stephenson's opener.
The 18-year-old forward, born in Blyth, latched onto a loose ball in the penalty area, despite Joe White looking to have been brought down inside the box beforehand, and slotted the ball beyond Leeds goalkeeper Dani van den Heuvel.
Elliot Anderson, a constant threat all afternoon for the hosts, went close to doubling Newcastle's lead after cleverly evading a challenge inside the penalty area but saw his low strike fly straight into the hands of the Leeds shot-stopper.
In the 43rd minute, Newcastle extended their advantage as Stephenson notched his second of the afternoon. After Nathan Carlyon's teasing through ball was directed into Lucas De Bolle's path, the midfielder's pass into the box was only cleared into Stephenson's path, who composed himself before drilling the ball home.
Despite rarely threatening Newcastle goalkeeper Max Thompson in the opening half, Leeds pulled a goal back as Snowdon connected with Ethan Kachosa's cross and tucked the ball into the far left-hand corner.
In the third minute of additional time, Newcastle regained their two-goal cushion as Stephenson completed a marvellous hat-trick. After Anderson's inviting cross was sent across the six-yard area, the teenage forward was on hand once again to turn the ball into the net.
In the second-half, the hosts began where they had left off as Anderson saw his dipping free-kick crash off the top of the net before the England under-19 international, who made his Premier League debut for the Magpies' first-team at Arsenal in January, then hit the post after his deflected effort from the edge of the box smashed off the underside of the left-hand post.
De Bolle, who has made four appearances for United's under-23s this term, went close to extending the young Magpies' lead as his powerful effort from the edge of the box was pushed away by van den Heuvel.
In the 64th minute, Newcastle added their fourth through White after the Carlisle-born midfielder, a representative of England at under-18 level, initially picked the ball up from distance before advancing towards goal, beyond the defender, before firing the ball into the back of the net.
Leeds pushed forward in the hope of a late fightback as substitute Ronnie McGrath saw his header crash off the crossbar but Newcastle comfortably saw out the game to seal a deserved place in the next round as the dreams of lifting the trophy for the first time since 1985 remained on course.
Newcastle United Under-18s: Max Thompson, Joe Oliver, Harry Barclay, Josh Nicholson, Josh Scott, Nathan Carlyon, Jamie Miley, Joe White (Kyle Crossley 74), Elliot Anderson (James Huntley 86), Lucas De Bolle (Reagan Thomson 83), Dylan Stephenson.
Subs not used: Josh Stewart, Piotr Banda, Josh Gilchrist, Josh Harrison.
