The Pilgrims trailed to a penalty from Chasetown's George Dixon, but hit straight back through a spot-kick of their own from Corey Newman.
Newman put the Pilgrims in front 22 seconds after half-time before Bradley Easey added a clinching third goal with seven minutes to play.
Boston (4-3-3): Travis Portas; Conor Allen, Billy Hodgkiss (Joe Money 50), Daniel Moulds, Jack McCallum (David Olalere 55); Miles Lynch (Liam Moran 50), Jack Dykes, Corey Newman; Steven Dovey, Scott Floyd, Bradley Easey. Subs (not used): Joe Cain, Karolis Andriulionis.