Evison struck in the 44th minute to settle a first qualifying round tie against Leicester Nirvana at The Jakemans Stadium.
The Pilgrims had an abundance of chances but could not add to their rally as they booked a place in Friday's second qualifying round draw.
Boston (4-3-3): Harry Payne; James Gordon, Callum Rodger, Bailey Forth, Jack Brown; Jordan Tate (Oli Drummond 66), Jack Withers, Harry Vince; Ben Lawrence, George Smith (Jenson Ball 88), Peter Evison (Connor Stocks 81). Sub (not used): Jack Henton.