The young Glovers knew victory would set up an away tie at Aston Villa or Blackburn Rovers but a dogged Boro side stood in their way.
Alie Bangura netted the game’s solitary goal and Boro, whose first team play in the ninth tier of English football, hung on to progress to the last 16.
With over 1,000 packed into Goffs Lane, fans of the Hertfordshire side craved an upset and were given early hope through Bangura.
The youngster netted in the opening exchanges to give the hosts a lead they did desperately well to hold on to.
Yeovil threw everything at their opponents in the first half as they searched for a way back into the contest, but found a resilient young Broxbourne defence in wait.
The second period produced a similar story as the Green and Whites found themselves on top with the home side trying to keep them at bay.
Town couldn’t find the crucial goal they needed and Broxbourne ground out their one-goal advantage to produce the shock of the round.