The hosts created the first opening after five minutes. Grant did well down the left and crossed, but James Ryan blazed a shot well over the crossbar from 12 yards.
Chesterfield responded with Jack Lester firing straight at keeper Tommy Lee but at the other end, Chesterfield defender Jared Wilson had to be alert to hack an effort from Craig Lindfield off the line.
Back came Chesterfield and midfielder Drew Talbot raced clear down the right but he crashed a shot high over the crossbar from just inside the box.
There was little to choose between the two teams in the opening half hour and at that point Stanley's Grant let fly with a thumping volley that whistled past the outside of a post.
The young midfielder followed that up with a low drive that flew just wide two minutes later.
On 35 minutes, Accrington winger Ryan produced an amazing run towards the Chesterfield back line but the shot at the end of it fizzed narrowly wide.
Chesterfield missed a great chance a minute before the break. Jamie Lowry crossed in but Lester sliced his shot, allowing Kenny Arthur to save.
The visitors started the second half purposefully and twice within a minute, Talbot fired efforts wide of the Stanley goal.
However, a superb goal opened the scoring. Peter Cavanagh played a long cross-field ball from the right-back position and Grant shot first time high into the top corner from 18 yards.
Alan Goodall headed wide as Chesterfield tried to hit back quickly, then substitute Jamie Winter saw a low shot well gathered by Arthur.
However, a dismal day got worse for the visitors when star frontman Lester was shown a straight red card for dissent.