The 20-year-old, on loan from npower Championship outfit Huddersfield, pounced on a loose ball in the 76th minute to condemn the Shots to a fourth straight defeat and leave the Hampshire club second from bottom of the table.
Earlier in the second half, Atkinson fired just wide while at the other end Shots striker Danny Hylton could not convert his chances.
Stand-in Chesterfield skipper Sam Togwell cleared a Troy Brown header off the line after three minutes before Hylton registered his first shot on target moments later.
Brown then almost headed Tendayi Darikwa's cross into his own net while at the other end Spireites goalkeeper Richard O'Donnell had to be alert to save a Alex Rodman header.
Togwell had a shot saved early in the second half and Atkinson hit a sweet volley from 25-yards wide of the left-hand upright. He made no mistake with 14 minutes remaining though, while Hylton could have grabbed an equaliser for the Shots at the death but his header drifted wide.