Mason Warren's fine strike on 59 minutes, following excellent build-up play from Harry Vince, ensured a safe passage for Adam Murray's men.
The Pilgrims will now face Grimsby Town or Scunthorpe United, at home, in the last four.
Boston (4-2-3-1): George Willis; Taron Hare (James Clifton 46), Jack Cowgill, Jordan Keane, Brad Beatson (Jan Yeomans 68); Jack Broadhead, Adam Chapman (Tom Batchelor 68); Tyrell Waite (Liam Adams 68), Ashley Hemmings (Harry Vince 46), Mason Warren (Kuda Muskwe 60); Kabongo Tshimanga (Darnelle Bailey-King 46).
Referee: David Jones.