Sam Harris had given the Pilgrims a fifth minute lead, only for Chesterfield to turn the tables with a quickfire double, but United led 3-2 at the break courtesy of two Johnson strikes.
The Sprieites, down to 10 men, equalised on the hour mark, but United came on strong during the closing stages, with Johnson adding two more goals, alongside efforts from substitutes Elliot Foster and Zane Millar. Chesterfield had briefly reduced the arrears to 6-4 before Johnson's final goal.
Boston (3-4-2-1): Charlie Wilkinson; Bailey Dilley (Taylor Jaine 60), Owen Sheriff, Ben Robson; Krisztian Gagyi (Zane Millar 55), Haydn Tear, Joe Melson (Mackenzie Burdass 70), Logan Tate (Elliot Foster 46); Finlay Armond, Ben Johnson; Sam Harris (Han Stevens 75).
Referee: Alistair Wilson.