Dan Senda’s side played with a clear confidence from the first whistle and nearly took the lead when Antonis Vasiliou flashed a shot narrowly wide from the edge of the box 7 minutes in.
Aymen Azaze comfortably claimed a free-kick in a dangerous position taken by Abdel Sahnoun.
Vasiliou came close again on the 15-minute mark, his fine left-footed effort striking the bar.
Okem Chime then had an acrobatic effort deflected over the bar before a free kick from Vasiliou forced a great save from the visiting keeper Jack Wilmot.
Vasiliou then turned provider, supplying crosses to both Chime and Joseph Taylor, but both efforts went narrowly over.
The Young Bees were awarded a free-kick shortly after the half time break and what followed look to come straight from the training ground.
Vasiliou and Martyn Box both stood over the ball and ran to take it, Vasiliou played it short to Rio Connell who cooly rolled his effort into the bottom right corner to give Barnet the lead.
That lead was doubled in the 63rd minute when Chime’s low effort was palmed straight into the path of substitute Dimeji Adeniran who tapped home from close range.
Box would be the next to look to get on the scoresheet, but his free kick from all of 30 yards was saved.
The ever-lively Vasiliou then turned three markers in one move on the edge of the box before finding Jack Tompkins who struck the crossbar.
The Young Bees controlled the final few minutes to secure the win and clean sheet and their place in the next round of the FA Youth Cup.
Barnet: Aymen Azaze, Dan Francis, Martyn Box, Jack Tompkins, Dylan Pauw, Moses Omore, Antonis Vasiliou, Joseph Taylor (Ronnie Edwards 87), Wraynel Hercules (Dimeji Adenrian 46), Okem Chime (George Murphy 75), Rio Connell.
Unused: Cian Moore, Fin Titchmarsh