The 19-year-old slotted home from the edge of the box after Barnet failed to clear Billy Kee's shot.
Stanley could have taken the lead before then, as a sliced clearance sent Michael Symes through but his skidding strike was parried clear by Bees goalkeeper Jake Cole.
Accrington midfielder James Ryan came close to doubling the lead minutes after Grant's opener but his dipping volley just shaved the outside of the post.
Barnet had a gilt-edged change to level on the stroke of half-time but neither striker Paul Furlong nor Plymouth Argyle loanee Yannick Bolasie could convert Albert Jarrett's cross.
The hosts threw men forward after interval but substitute Dean Sinclair's headed inches over the bar, while at the other end Cole denied Symes late on.