Easey struck twice to sink Matlock Town at the Autoworld Arena and book a trip to Basford United in the next round.
United had carved out plenty of opportunities before eventually breaking the deadlock, with Easey at the forefront of their attacking play, while captain Daniel Worthington hit the post with a free-kick.
Boston (4-3-3): Jamie Bradley; Daniel Worthington, Billy Hodgkiss (Alex Dempster 74), Joe Cain, Jack McCallum; Miles Lynch, Liam Moran, Corey Newman; Dominic Goddard, Louis Saebe (Jack Dykes 82), Bradley Easey. Subs (not used): Jack Foyster, Scott Floyd, Joe Money.
Referee: Martin Beard.