In a match of few clear-cut chances, Michael Cornick’s left side cross was sliced past Jamie Stephens by the Barnet defender to give Yeovil the lead on 37 minutes.
The Bees went into the break a goal down, but it was Barnet who had the better of the first half.
After 16 minutes Aaron McLean's came close to opening the scoring - firing a direct free kick narrowly over the bar.
McLean was again involved in the action four minutes later; flicking on a Sam Muggleton long throw to Dembele, whose header was high and off target.
On 26 minutes a John Akinde pass found Curtis Weston on the edge of the box, but his shot went sailing over the crossbar.
McLean connected with another Muggleton throw into the box just before half time, shooting wide of the left post by some margin.
In the second half Barnet’s best chance fell to Weston – the midfielder’s 56th minute long shot flew wide and didn’t trouble Glovers keeper Artur Krysiak.
It wasn’t until the 68th minute that Yeovil had their first real effort of the match; a low Matthew Dolan shot from distance went wide of the left post.
As Barnet pushed forward, Yeovil created two further openings.
On 79 minutes Shaun Jeffers danced into the Bees’ box, before shooting over from a good central position.
Then after 83 minutes substitute Jordan Gibbons let fly with a right sided shot that went narrowly over the bar.
Barnet pushed for an equaliser and even threw Stephens up for a corner in the final minutes of the match.
The Bees couldn’t find a way through the Yeovil defence though, and it is the Glovers who go in the hat for the 2nd round.
BARNET: Stephens, Hoyte, Dembele, Johnson (Muggleton 7’), Nelson, Weston, Akinde (Stevens 45’), Tomlinson, Champion, Batt (Gondoh 55’), McLean.
Subs not used: N’Gala, Gambin.