Lucas De Bolle's composed finish inside five minutes gave the young Magpies a perfect start at St. James' Park before Will Hall's strike drew the Hornets level.
Dylan Stephenson, who netted a hat-trick in a 4-1 win against Leeds United in round four, continued his goalscoring run in the competition with a clinical finish before Newcastle's teenagers were pegged back in the second-half through Tiago Cukur's penalty.
With penalties approaching after a gruelling 120 minutes, Anderson secured United's passage into the last eight with a delightful stoppage-time free-kick in front of the Gallowgate End to complete a five-goal thriller.
Neil Winskill's side, who had beaten Huddersfield Town and Leeds in the previous rounds, got off to a wonderful start in the 5th minute through De Bolle's strike.
After Anderson capitalised on a defensive error, the ball fell into Stephenson's path who cleverly picked out De Bolle inside the penalty area, who calmly tucked the ball home.
Bosun Lawal went close to netting for the visitors as the Hornets skipper headed over the crossbar from Sonny Blu Lo-Everton's swinging corner.
In the 18th minute, Watford equalised through Hall as Blu Lo-Everton's looping free-kick was headed towards goal by Lawal before Shaq Forde made contact with goalkeeper Max Thompson.
Despite the teenage shot-stopper being left on the floor after a nasty collision, play was not halted, allowing Hall to place the ball into an empty net.
After a negative blow, Newcastle regained their lead in the 33rd minute as Stephenson netted his fourth goal in two Youth Cup matches.
After clever footwork from De Bolle to advance past a pair of Watford defenders on the left wing, the midfielder threaded a brilliant through ball into the striker's path, who drilled a fierce first-time effort past Watford goalkeeper Alfie Marriott.
The 18-year-old almost gave the hosts a two-goal advantage before the half-time break after getting onto the end of Josh Scott's teasing low cross, following Joe White's neat pass into the left-back's path, but saw his strike remarkably saved from point-blank range by Marriott.
In the second-half, White - who scored in last week's 4-1 win against Leeds - went close to netting in successive matches shortly after the restart but his effort from the edge of the box was saved by Marriott.
Thompson then made a pair of important stops for the young Magpies, first denying Forde's goal bound strike before producing an equally impressive save a minute later, pushing away Blu Lo-Everton's stinging drive.
Stephenson was denied again by Marriott as the Hornets shot-stopper tipped the forward's powerful header out for a corner following Jamie Miley's inviting cross from the right flank.
In the 76th minute, Newcastle found themselves on level terms for a second time as Josh Nicholson tugged down Forde on the byline, with Cukur dispatching the resulting spot-kick in emphatic fashion.
In extra-time, both sides pushed for an all-important third goal, with De Bolle unable to notch his brace after firing over in the opening 15 minutes.
In the dying stages after two hours of play, Anderson delivered United's third goal of the afternoon in incredible fashion.
After Watford conceded a free-kick on the edge of the penalty area, the England under-19 international lifted the ball over the wall and into the far left-hand corner to book a quarter-final tie at home to either Aston Villa or Burton Albion.
Newcastle United Under-18s: Max Thompson, Joe Oliver, Harry Barclay, Josh Nicholson (Kyle Crossley 98), Josh Scott, Nathan Carlyon, Jamie Miley (Reagan Thomson 106), Joe White, Lucas De Bolle, Elliot Anderson, Dylan Stephenson (James Huntley 120).
Subs not used: Josh Stewart, Piotr Banda, Lewis Brannen, Michael Ndiweni.