Grant was widely linked with a move away from the Crown Ground last month and further underlined his potential, netting his 12th goal of the season after 17 minutes of the second-round replay at Underhill.
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.