Playing with greater thrust and belief, Stevenage were rewarded only six minutes after the restart when Luke Oliver nodded the ball down to Cole, who finished smartly from the right-hand side of the box.
The midfielder, who plays for Turkish club Genclerbirligi, headed home after Korea failed to clear a corner from Luke Wilkshire and Lucas Neill nodded the ball back across the goal.
Forest twice almost levelled when Morgan nodded over and then Luke Steele saved well from Earnshaw, but it was Barnsley who scored again when Trippier got away down the right and outfoxed Cohen before crossing for Gray to head home.