The hosts flew out of the traps, a flurry of crosses were half cleared by the Bees and in the 7th minute Keshi Anderson had a shot deflected behind.
The Bees took the lead 9 minutes in following a defensive mix up. Chris Robertson and ‘keeper Lawrence Vigouroux left the ball to one another which allowed Shaq Coulthirst to nip in and tap home.
Just over 5 minutes later it was 2 as Ricardo Santos was hauled down in the area and the Bees were awarded a spot kick.
Jamal Campbell-Ryce stepped up and placed it low into the bottom right hand corner.
Debutant Dave Tarpey had a chance blocked by a Swindon man in the 24th minute, Andre Blackman’s follow up was also blocked.
Early into the second half, the hosts were reduced to ten after Donal McDermott saw red for a very late tackle on David Tutonda.
Swindon continued to press, Craig Ross pulling off a fantastic save to deny a bullet header from Robinson.
Barnet had a golden chance on the counter attack, Jack Taylor sent through Dave Tarpey but his looping effort cannoned back off of the bar.
Moments later, Ross was called into action again, this time getting down low to save a Keshi Anderson shot.
In the 64th minute, Shaq Coulthirst got his second after JCR set him up one-on-one with the ‘keeper and he made no mistake.
Five minutes later he got his hat-trick. Good passing play in the build-up found Coulthirst who twisted and turned to find space and sumptuously curl a shot left footed past Vigouroux.
Swindon pulled a goal back in the 77th minute, Luke Norris rising highest to head past Craig Ross.
Two minutes later, Barnet were reduced to 10 – Andre Blackman dismissed for an off the ball incident spotted by the fourth official.
Barnet would lose one more player in stoppage time, Ricardo Santos stretchered off.
Barnet: Ross, Santos, Nelson, Campbell-Ryce, Coulthirst (Mason-Clark 87’), J. Taylor (Fonguck 90’), Tutonda, H. Taylor, Vilhete, Blackman, Tarpey (Akpa Akpro 59’)
Unused: R. McKenzie-Lyle, Clough, Bover, Amaluzor.