After a tight encounter at the Rush Green Training Ground, the third round tie sprung into life in the closing stages as Michael Ndiweni's close-range rebound broke the deadlock for the Toon teenagers before the visitors were quickly pegged back through Divin Mubama's emphatic finish.
A dramatic finale was to follow as Jay Turner-Cooke regained United's lead in the dying minutes before Ndwieni's composed finish eventually wrapped up the victory for Newcastle's youngsters despite George Earthy setting up a nervy finish after halving the deficit for the hosts.
Newcastle's win sets up a fourth round tie at home to Colchester United, who claimed a 3-0 victory against Arsenal.
The young Magpies, managed by former United centre-back Peter Ramage, reached the quarter-finals of the prestigious youth competition last term before losing to eventual winners Aston Villa.
Turner-Cooke created the opening goalscoring opportunity of the cup tie as the 17-year-old winger, who joined the Magpies from Sunderland in January, latched onto a headed clearance before his drilled strike towards goal was blocked by Hammers defender Kaelan Casey.
United goalkeeper Max Thompson, who made his England under-18s debut in October, was forced into action shortly after as the teenage shot-stopper held Earthy's vicious drive from the edge of the penalty area.
Newcastle, who have lifted the Youth Cup trophy on two occasions - in 1962 and 1985 - headed into the half-time break on level terms despite Mubama heading narrowly wide from Regan Clayton's in-swinging cross.
In the second-half, United pushed for an opener as Turner-Cooke's free-kick flew over the crossbar before Kyle Crossley - who scored on his debut for Newcastle's under-23s against Crystal Palace in April, saw a mazy run thwarted by Casey.
In the 70th minute, Newcastle opened the scoring as Ndiweni registered his 11th goal of the season at under-18 level in all competitions.
As Crossley intercepted Freddie Potts' pass, the teenage winger advanced forward picking out Ndiweni inside the box and after seeing his first attempt blocked by West Ham goalkeeper Jacob Knightbridge, the 18-year-old forward slotted the ball home from a tight angle.
Newcastle's lead lasted five minutes, though, as West Ham equalised through Mubama's clinical strike. As Junior Robinson's cross fell kindly into the feet of the forward, the 17-year-old lashed the ball past Thompson.
With extra-time looming, Newcastle found themselves ahead in the 89th minute through Turner-Cooke, who tucked the ball beyond Knightbridge after James Huntley's initial shot was deflected into the youngster's path.
A late flourish of goals were to arrive as Ndwieni notched his second of the evening in stoppage-time. After Huntley advanced down the right wing before cutting inside, the teenager picked out Ndiweni inside the box, who calmly evaded a sliding challenge before burying his effort into the net.
West Ham set up a tense finish as Earthy scored eleven seconds after the kick-off as Lewis Orford latched onto Knightbridge's long ball before teeing up Earthy, who placed the ball into the net.
However, Newcastle held firm in the final seconds to claim a hard-fought victory in the capital and book their spot in round four.
Newcastle United Under-18s: Max Thompson, Josh Stewart, Ciaran Thompson, Harry Barclay (Callum McNally 24), Beau Beresford, Ellis Stanton, Jamie Miley (c), James Huntley, Kyle Crossley, Jay Turner-Cooke (Josh Donaldson 90), Michael Ndiweni.
Subs not used: Steven Bessent, Lewis Miley, Dylan Charlton, Ben Parkinson, Shaun Mavididi.
