Ben Parkinson's header broke the deadlock for the young Magpies in the latter stages of the first-half before Olly Sanderson's low drive drew the Bantams level before the half-time break at the University of Bradford Stadium.
United's youngsters fell behind in the second-half following Andy Cook's close-range finish before Josh Donaldson's stoppage-time effort, flicking the ball home from Jay Turner-Cooke's free-kick, sent the contest to a penalty shoot-out.
Despite Ellis Stanton being denied in the shoot-out, Magpies goalkeeper Aidan Harris saved spot-kicks from Alex Pattison and Cook as the young Magpies prevailed on penalties following successful conversions from Parkinson, Turner-Cooke, James Huntley and Donaldson, resulting in Newcastle's first win in the competition since 2018.
With a number of Magpies youngsters away on international duty at youth level, including the likes of Alex Murphy, Trevan Sanusi and Travis Hernes, four changes were made to the side which suffered a 1-0 defeat against Derby County on Friday as Johnny Emerson, Alfie Harrison, Dylan Charlton and Charlie McArthur were all handed starts against the Bantams.
Harris, who signed his first professional contract last summer, was forced into action in the opening stages as the 17-year-old shot-stopper pushed away Joe Adams' low cross-cum shot.
In front of 1,790 spectators, Harrison went agonisingly close to opening the scoring as the Keighley-born teenager, who made his full senior debut in the Magpies' Sela Cup victory against Stade Brestois last month, saw his curling effort from the edge of the box strike the foot of the left-hand post.
In the 40th minute, Newcastle took the lead as Parkinson completed a fine attacking move, guiding a powerful header from Johnny Emerson's cross past Bradford goalkeeper Colin Doyle.
The League Two side, who made eight changes from last weekend's defeat against Grimsby Town, drew level two minutes later through Sanderson as the City forward drilled a fierce low effort into the bottom left-hand corner.
Sanderson looked to have netted his and Bradford's second goal of the night on the stroke of half-time after guiding a header into the net from Brad Halliday's cross from the right flank but the striker was penalised following a push on Magpies skipper Cathal Heffernan.
Jay Turner-Cooke went close to regaining United's lead shortly after the hour mark as the South Shields-born youngster, who joined the Magpies' youth set-up in January 2021, latched onto a loose ball inside the penalty area before his venomous drive at goal was turned over the crossbar by Doyle.
In the 76th minute, Bradford took the lead through Cook as the City substitute, who scored 28 League Two goals last season, was on hand to tap the ball home from close-range following Jamie Walker's dangerous cross into the six-yard area.
Harris produced a strong stop to deny Walker from doubling the hosts' advantage, with the teenage 'keeper getting across quickly to push away the midfielder's teasing free-kick from the edge of the box.
In the second minute of stoppage-time, United drew level through Donaldson as the Gateshead-born striker was on hand to inadvertently flick Turner-Cooke's low free-kick into the net from close-range.
In the shoot-out, Parkinson converted United's first penalty before Stanton's attempt, directed to the left-hand corner from 12 yards, was saved by Doyle.
Turner-Cooke, Huntley and Donaldson, however, kept United's hopes of an additional point alive with successful penalties before Harris proved to be the hero, with the young goalkeeper denying Pattison from the spot after guessing the correct way before then blocking Cook's penalty.
Newcastle United Under-21s: Aidan Harris, Ciaran Thompson, Cathal Heffernan (c), Charlie McArthur, Dylan Charlton, Ellis Stanton, Anthony Munda (James Huntley 75), Jay Turner-Cooke, Alfie Harrison (Josh Donaldson 60), Johnny Emerson (Guy Bloomer 88), Ben Parkinson.
Subs not used: Adam Harrison, Scott Bailey, Harry Powell, Rory Powell.
