The striker converted his eighth of the campaign from the penalty spot early on after skilfully winning a penalty against Ricardo Almeida Santos.
Jonny Maddison made some outstanding saves in the second half to preserve the Glovers’ lead and Zoko scored late on to confirm the win.
Darren Way made five changes from the side who drew with Crewe Alexandra on the weekend, two of which were enforced.
Artur Krysiak and Nathan Smith failed to recover from knocks picked up at Gresty Road and were replaced by Maddison and Ryan Dickson – the former making his first start of the campaign.
A new strike pairing was on show as Surridge partnered Alex Fisher up front while Oscar Gobern appeared in green and white for the first time since injuring his hamstring in December.
Referee James Linington deemed the pitch playable following an earlier inspection and Town proved his decision was justified inside a minute.
It was just like watching Brazil as the hosts passed the ball about with slickness and Dickson’s excellent cross was beautifully taken down in the area by Surridge before being hauled down by Santos.
The Bournemouth loanee dusted himself off and finished emphatically from 12 yards to give Yeovil the perfect start.
After a frantic beginning, proceedings quietened with Simeon Akinola curling over for the visitors for the only other chance in the opening 15 minutes.
The forward duo of Surridge and Fisher looked to stretch Barnet at every given opportunity and the latter’s fizzing delivery across the corridor of uncertainty almost brought a second home goal.
Charlie Clough cleared and an onrushing Jake Gray’s shot was well stopped by Craig Ross before Jordan Green went down in the box, however nothing was given despite appeals from the Thatchers terrace.
Tom James found the target with a trademark free kick from an acute angle and Surridge clipped the outside of the post after good play from Corey Whelan as the half eased to a conclusion.
Before it did, Richard Brindley’s powerful drive whistled past the post as a warning to let the Glovers know there was another team in the contest despite their dominance.
The second half nearly began in similar fashion to the first, this time Barnet doing the attacking.
Akinola’s low attempt was deflected towards the bottom corner and a superb reaction stop from Maddison prevented an equaliser.
Full-back Richard Brindley and midfielder Alex Nicholls saw further attempts fly just over as Yeovil struggled to gain control as they had done in the first period.
Then Ryan Watson then capitalised on a mistake in the home ranks and when his shot was expertly tipped away by Maddison, James needed to be quick to clear ahead of an onrushing John Akinde.
Gray forced Ross into a flying stop on 70 minutes as Town pressed for an all important second and sweet strike from Whelan looked destined for the back of the net but for an unbelievable goal-line clearance by Santos.
Maddison was equally as good at the other end to deny substitute Mauro Vilhete, getting a fingertip to push the ball on to the post and then pushing David Tutonda’s drive wide.
The Durham-born stopper palmed over Brindley’s fierce shot soon after and James cleared off the line as the lowly Bees peppered the ball at goal.
Zoko netted at the near post to seal the game in injury time but the three points were largely down to the fantastic Maddison who deserves great credit for his display.
Yeovil Town: Maddison, Whelan, James, Sowunmi, Dickson, Gray, Gobern (Santos 80), Wing, Green (Browne 58), Fisher (Zoko 52), Surridge.
Bookings: Zoko 65.
Not Used: Phillips, Donnellan, Bird, Seager.
Barnet: Ross, Brindley, R.Santos, Clough, Tutonda (Blackman 84), Nicholson (Coulthirst 50), Weston, Watson, Akinola (Vilhete 72), Nicholls, Akinde.
Bookings: Brindley 49.
Not Used: Legg, Plavotic, Sule, Taylor.