The winner came in controversial circumstances as referee Carl Berry pointed to the spot, having adjudged David Stephens to have handled Sonny Bradley's goal-bound drive.
Craig Reid stepped up and confidently slammed the spot kick past Graham Stack, and things went from bad to worse for the hosts as they were reduced to ten men when Jon Nurse received his marching orders for a dangerous challenge on Dani Lopez.
And Bees boss Edgar Davids was enraged yet further as his men were denied a leveller, with Jake Hyde ruled offside having bundled home Ricky Holmes' near-post centre.
"It's a massive win for us today. I'm not going to apologise for pushing the players and striving for better," said Holdsworth.
"With the standard of the players in that dressing room, we want to be pushing up the division and it's really important that we do that.
"It wasn't going to be a classic game. Both teams are working extremely hard for their survival and wins. On the basis of that, to come here against a Barnet side that has been in confident mood is a fantastic win.
"It was handball - I don't think anyone does it intentionally now, but you can't have the ball touching your hand in the penalty area. My own personal opinion is that there wasn't intent.
"It looked, with the momentum Jon [Nurse] had going in on Daniel, that it could have been reckless. The referee hasn't got a rewind, but it looked quite a bad challenge."