Dembele sliced a Harry Cornick cross into his own net on 38 minutes to give the Glovers a lead but in truth neither side forced the opposing keeper into much work. Aaron McLean hit the crossbar in the first half but despite the game being more open late on, chances were at a premium but most importantly Yeovil make the second round draw and taste victory for the second time this season.
Yeovil boss Paul Sturrock made two changes to the team that were well beaten at Oxford United on Saturday. Jakub Sokolik dropped out of the sixteen completely and Ryan Bird took a bench spot with Shaun Jeffers coming in up front to play alongside the unlikely figure of Omar Sowunmi.
Artur Krysiak was the first of the keepers called into action when he had to get down smartly to deny Aaron McLean's near post effort. Bees boss Martin Allen was then forced into an early change when Elliott Johnson limped off and was replaced by Sam Muggleton.
McLean then came inches away from giving the visitors the lead on 17 minutes. The former Hull City striker smashed a free kick from fully 35 yards but it glanced off the top of the crossbar with Krysiak beaten. Bira Dembele then came close with a header that dropped on to the roof of the net before Bees captain Curtis Weston turned and shot way over.
On 38 minutes, the moment of farce that decided the game. Harry Cornick swung in a right foot in swinging cross that seemed to elude everyone and end up in the net but on closer examination, Barnet centre half Dembele got an awful connection to the cross and diverted the ball past keeper Jamie Stephens.
The half closed with McLean pulling a shot wide from a long Muggleton throw. In fact, Barnet's best hope of an equaliser lay with the prodigious rows of Muggleton who was reaching the penalty area from near the halfway line and the tactic was well used in the second period.
As the game progressed, it got more open but scrappier too with both sides giving away possession way too often. Ref Tim Robinson booked four players in a quarter of an hour, none of which were nasty fouls, more cynical than anything but it summed up the nature of the match.
One bright spark of life came through Jeffers who showed a clean pair of heels to his marking defender but fired way over from a tight angle. Glovers sub Jordan Gibbons looked lively and threatening and almost grabbed a late second with a bicycle kick that flew wide.
That was the final chance of a poor game but no matter, it's Paul Sturrock's Yeovil team that will be in Saturday's second round draw.