Adams headed home James Gordon's cross with 11 minutes to play to settle a tightly-contested encounter against Chasetown.
Jack Withers' first half header had earlier been cancelled out by James McGrady.
Boston (4-3-3): Harry Payne; James Gordon, Jack Withers, Callum Rodger, Jack Brown; Liam Adams, George Hobbins, Peter Evison (Connor Stocks 46); Ben Lawrence (Jenson Ball 82), Oli Drummond, Haydn Havard. Subs (not used): Jordan Tate, Arkadiusz Michalczyk.