The experienced midfielder, 35, was released by the Spireites in the summer after four years at the club.
Weston, who was a regular starter when Rowe was in charge at Town, will now play under him again at the National League North side.
He has penned a one-year deal and will officially become a Coasters player on July 1.
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Fellow ex-Spireite, Joe Rowley, who was also let go at the end of last season, joined Fylde earlier this month.
On signing Weston, Rowe said: “Curtis is a player that I have a strong relationship with, and I have worked closely and successfully with in the past. He will bring many on and off-field attributes to push the club forwards.”
And Weston said: “It’s been a deal we’ve been talking about for a few weeks now.
“I’ve met all the lads and had a little look around the place, and it looks incredible, so I’m ready to go.”