Newsham, back from the World University Games, fired Boston into a 12th minute lead, only for Lewis Cook to equalise for the Whites (managed by ex-Pilgrim Neil Redfearn) nine minutes later.
The match appeared to be heading for a stalemate until trialist Billy Bennett struck the winner for Boston in the 88th minute.
Boston (4-4-2): Ashley Timms (Sam Vince 46); Conor Marshall, Scott Garner, Stefan Galinski (Carl Piergianni 46), Netan Sansara (Spencer Weir-Daley 46); Ben Fairclough (Billy Bennett 46), Ian Ross (James Reed 46), Gary Mills (Ben Milnes 62), Ricky Miller; Charley Sanders (Jordan Nuttell 46), Marc Newsham.
Referee: Wayne Porter