The Addicks took the lead in the second minute of the game when Ahearne-Grant latched on to a through ball from Jordan Cousins and fired the ball past the keeper.
Ahearne-Grant notched his second strike of the game minutes later as he raced on to a long ball over the top of the Welling defence and lobbed the ball over the keeper and into the net.
The forward nearly notched his hat-trick when he met a cross into the box and powered a header at goal, but his effort was cleared off the line.
In the second half, Mikhail Kennedy came close to extending his side's advantage but he was denied by a fine save from the Welling keeper.
Welling upped the ante late in the game as they struck the post, while player-manager Loui Fazakerley hit the bar with a powerful volley, but the Addicks were able to see out the win.