Substitute Millar Matthews-Lewis hit a superb second-half hat-trick after coming on just before the break, after Conor Lawless and Jake Burger had given the development squad a perfect start with two stunning goals.
Avan Jones netted the other from the penalty spot, with Biggleswade replying through Alex Marsh and captain Tom Coles with the last kick of the game.
The young Hatters started like a train, Bim Pepple and captain Casey Pettit drawing saves from Dan Green from distance inside the opening three minutes before the Biggleswade keeper had to acrobatically push a deflection from his own centre-half Dan Newton away with just five on the clock.
Lawless broke the deadlock in the eighth minute with a superb 25-yard shot into the top corner after Biggleswade had only half-cleared a ball into the box.
It was almost two moments later when Pepple again found space on the edge of the box, but dragged his right-footed effort wide of Green's near post.
Town were playing some cracking football and Jayden Luker was the next to call Green into action midway through the half, bursting into the box and forcing the Biggleswade number one to save with his feet by his near post.
The lead was doubled just before the half-hour by another cracker, this time from Burger, who collected the ball just outside the area after Joe Johnson had done brilliantly to dance his way to the left byline, and bent another beauty into the opposite top corner.
James Shea had to be alert in the 36th minute when Biggleswade wideman Alex Marsh broke free and forced the Town No.1 into a fine diving save to tip the ball around his right post, but Burger was soon on the hunt at the other end, slipping a shot just past the post after racing through the heart of the visitors' defence with just Green to beat.
Biggleswade were growing into the game and sounded another warning three minutes before the break when Coles was off target with a shot from just inside the box, but the Hatters could have made it three seconds later when Oli Lynch had his first sight of goal, only for his angled effort to go wide of the far post.
The Southern Leaguers were enjoying a fair share of possession at the start of the second half when Coles climbed highest to meet a right-wing corner from Charlie Hayford, only to direct it straight at Shea.
Biut the Hatters had soon extended their lead when Luker crossed from the left for sub Matthews-Lewis, on for Pepple a minute before the break, to steer his effort into the bottom corner.
Green saved well again to deny Lynch a fourth on 64 minutes following an incisive passing move down the Town left involving Johnson, Luker and Lawless, but Biggleswade wouldn't lie down, and substitute Reiss Oteng was soon feeding Michael Simpson to turn and take aim at Shea's goal, but narrowly off target.
Coles headed Oteng's excellent left-wing cross over with just under 20 minutes to go, but the Hatters were handed the chance to make it four in the 74th minute when Luker was tripped by Oteng in the box and Jones stepped up to send Green the wrong way from 12 yards.
Still Biggleswade kept coming, Coles bursting through the middle to hit a first-time effort just over in the 79th minute, then they were handed a gift with seven minutes remaining when Alex Marsh capitalised on a loose backpass from Ben Tompkins to beat Shea in a foot-race and tap home.
Town's four-goal advantage was restored within a minute however when Matthews-Lewis grabbed his second from Lynch's cross, via a Casey Pettit flick, before he completed his hat-trick with 90 minutes up with a lovely lob over Green after latching onto Jones' through ball.
There was still time for Biggleswade's outstanding skipper Coles to hit a 20-yard free-kick past the Hatters' wall with the final kick of the game, but it was the young Hatters who would lift the trophy in front of the Enclosure after a brilliant display.
TOWN: Shea, Jones, Tompkins, Francis-Clarke, Johnson, Lawless, Pettit (C), Luker, Burger, Pepple (Matthews-Lewis 44') (Piesold 90'), Lynch.
Subs: Bateson, Pinnington, Horlick.
Goals: Lawless 8', Burger 29', Matthews-Lewis 56', 84', 90', Jones (pen) 74'
BIGGLESWADE: Green, Hunt, Simpson, Akubuine (Wedd 61'), Newton, L Marsh, A Marsh, Dreyer, Bond, Coles, Hayford (Oteng 61').
Subs: Wedd, Angol, Bailey.
Goals: Marsh 83', Coles 90+4
REFEREE: Mario Stetakovic
ATT: 751
