Asante volleyed home from inside the area to give the Spireites the lead on nine minutes.
Luke Young equalised from the spot after Jak McCourt was harshly adjudged to have handled the ball.
But with the clock ticking down, Asante was put through on goal by Liam Mandeville and he slotted in the winner and his sixth in a Town shirt.
Here are Liam Norcliffe’s player ratings….
1. Adam Przybek 7
The 20-year-old, who signed on loan from Ipswich Town this week, made his debut. He did not have any big saves of note to make. His distribution was good, he punched a couple of corners clear and commanded his box well. Photo: Tina Jenner
2. George Carline 7
Solid. Kept it simple, strong in the tackle and his long throws were a threat all afternoon. Moved over to left wing-back late on and slotted in nicely. Photo: Phil Tooley
3. Will Evans 7
The skipper was fairly solid defensively. His distribution could have been a bit better as he went long with his passing a few too many times, but some down the channels did work. Photo: Dean Atkins
4. Haydn Hollis 7
His good form continues. Played it out from the back and was very vocal throughout. Can't fault him. Had a strong shout for a penalty in the first half when his flicked header appeared to hit a Wrexham hand. Photo: Dean Atkins