Bradley Easey was on target for the Pilgrims, but the Derbyshire outfit proved too strong.
Goran Seymour grabbed a hat-trick, while Ryan Osborne netted Ilkeston's other goal.
Boston (4-3-3): Jamie Bradley; Daniel Worthington, Josh Spencer, Ollie Price, Corey Newman; Miles Lynch (Steven Dovey 60), Jacob Pease, Aaron Eyett; Leon Squire, Liam Moran (Karolis Andriulionis 60), Bradley Easey.