Braces for Danny Newton and Alex Revell secure victory for the hosts, Shaq Coulthirst's solo effort only a consolation.
It was the Bees that started the match on the front foot, Ryan Watson hitting the crossbar with an in-swinging corner within the opening 10 minutes.
A minute later, Joe Payne, making his EFL debut on his 19th birthday, burst down the left, nutmegged his marker before finding Jean-Louis Akpa Akpro who delivered a dangerous cross towards the far post which a Stevenage defender had to head away from under the crossbar.
Midway through the first half, Payne delivered a cross towards John Akinde but Tom King just beat the Barnet number 9 to the ball.
Soon after, Akpa Akpro drove forward from the halfway line and opted to go alone despite the support of Akinde but his shot was off target.
The hosts had a chance to take the lead in the 38th minute when an effort at goal took a wicked deflection but Craig Ross did brilliantly to quickly palm away the effort.
Stevenage took the lead just a minute before the break, Newton getting on the end of Ben Kennedy's cross to head home at the far post.
After the restart, Curtis Weston had a chance to level but his effort from the edge of the penalty area went narrowly wide.
Newton then hit an unstoppable volley from 25 yards out to double the hosts advantage shortly after the hour mark.
Revell made it 3-0 five minutes later when he beat Craig Ross with a low effort.
With 15 minutes left to play, Coulthirst flicked on a Ryan Watson free-kick and forced a good save from kick away to his left.
Shaquile Coulthirst pulled one back with a great solo run before hitting a left-footed effort across the face of goal into the bottom corner giving Tom King no chance.
As Barnet pressed, Stevenage made it four on the counter when Newton raced down on goal before playing in Revell to secure the points.
Attendance - 3350 (942 Barnet fans)
Barnet: Ross, Santos, Nelson (J. Taylor 63'), Watson, Weston, Akinde, Coulthirst, Nicholls (Akinola 51'), Akpa Akpro, Sweeney, Payne (Tutonda 69')
Unused: K. McKenzie-Lyle, Brindley, Clough, Shomotun.