The Pilgrims' senior youth side chalked up their opening win - following successive 2-2 draws at the start of the season.
Horne's 17th minute penalty was doubled in the 58th minute - and he completed his treble nine minutes from time after Harvey Limb had briefly reduced the arrears for the Spireites.
Boston: Tyler Griffin, Ellis Perry-Burdett, Josh Doran, Rueben Fogg, Finley Ging, Taylor Jaine, Jake Henderson (Will Langford 63), Kris Zaluzinskis, Joe Wilkinson (Rhys Smaller 54), JJ Ward, Luke Horne (Brenden Parker 83). Sub (not used): Charlie Brooks.
Referee: Trevor Basker.