The hosts swept into a two-goal lead inside the opening eight minutes. Luke Foster opened the scoring when he tapped home from close range following an excellent low cross from Stacy Long.
Another cross was then diverted into his own goal by Grant Basey to double Stevenage's advantage. Long deservedly got onto the scoresheet himself just before the half-hour mark as he took a pass from Robert Sinclair before burying a shot low into the corner.
And within a minute of the restart Barnet were four goals down as Chris Beardsley played in Mark Roberts, and he made no mistake from an acute angle.
To their credit Barnet battled back and Izale McLeod scored from 12 yards, before Mark Marshall made it 4-2 with 14 minutes left.
Stevenage almost scored a fifth in the dying minutes when Chris Beardsley's thumping drive struck a post.