Adrian Forbes' youngsters scored a goal in each half, with Millar Matthews-Lewis netting the opener in the 24th minute after a cross from captain Jake Burger deflected off a defender to the far post, where the forward was waiting to volley home.
The Southern League Division One Central hosts had Shane Bush sent off for bringing down Josh Allen when the Town striker was through on goal six minutes before half-time.
Joshua Phillips added the second with 15 minutes remaining, getting on the end of a corner and finishing from close range, to ensure Town move into the third round with victory over the side they beat in the semi-final last season.
Forbes on the win at Bedford: "The boys played well against a tough team that were physical which we aren't used to.
"We passed the ball around excellently at times, we could have been a bit more clinical in-front of goal but we just didn't take those chances.
"The most important thing was making sure we got into the next round, which we did. The key for me was not wanting a different team playing in a cup final at Kenilworth Road other than us, which is what I said to the boys before the last game and before tonight."
TOWN: Horlick, Pinnington, Benagr (Kayibanda 85), Piesold, Tompkins, Francis-Clarke, Heron, Burger (C), Allen (Anderson 88), Phillips, Matthews-Lewis (Braney 85).
Subs not used: Camis
Att: 194