The Pilgrims trailed 1-0 at Raunds' Manor School Sports College but turned the match on its head after the break to win 5-1.
In addition to Adams and Vince each scoring twice, Haydn Havard also netted for the Pilgrims.
Boston (4-3-3): Harry Payne; James Gordon, Jack Withers, Bailey Forth, Callum Rodger; Peter Evison (George Hobbins 46), Harry Vince, Haydn Havard; Ben Lawrence (Connor Stocks 80), Liam Adams, Jack Brown (Jordan Tate 46).