The young Spireites entered the fixture off the back of a strong start to the season, having won their last two games 1-0 away at Stockport and Mansfield respectively.
Walsall were also in good form and crossed the midlands in hope of adding their fourth league win of the campaign.
The Spireites started the brighter and opened the scoring after just ten minutes through Vegari, who capitalised on Walsall’s failure to clear from a corner and swept home from inside the box for his first of the season.
Walsall pressed for an equaliser and came close after 28 minutes through an in-swinging set piece from the left, but Ashton Rinaldo was able to watch on as a headed effort flew over the bar.
The Spireites had a set piece themselves two minutes later, but Tom Pawson couldn’t divert goalwards from Barnaby Smith’s cross.
After Rinaldo denied Walsall at one end, a counterattack saw Vegari turn the defender and lay off Brassey who couldn’t find the target.
Brassey dragged another chance wide a minute later from the right side of the box.
Hollis’ side should have doubled their lead a minute before the break when Vegari was played in on goal, but the striker couldn’t get a clean connection on the ball and dragged his shot into the arms of the Walsall keeper.
After entering the field of play at half time, Elliott made his impact on the game and doubled his side’s lead with 13 minutes to go.
Controlling on the edge of the box, he smashed an effort over the Walsall keeper into the back of the net.
Elliott put the game to bed eight minutes later when his pressure on the Walsall stopper paid off, blocking his clearance and leaving an empty net to tap home for 3-0.
Spireites: Rinaldo, Smith, Sen (Wainman 76), Reah, Bointon, Whitney, Gill (Shone 81), McKernan, Vergari (Elliott 51), Pawson (Hague 59), Brassey (White 73)
