The Pilgrims trailed Blyth Spartans at Wensleydale Park, before equalising through Owen Sheriff.
With the match seemingly heading to a draw, United pinched all three points when Johnson struck two minutes into added time.
Boston (3-4-2-1): Charlie Wilkinson; Sam Harris, Owen Sheriff, Taylor Jaine; Logan Tate, Joe Melson, Mackenzie Burdass, Zane Millar; Finlay Armond, Ben Johnson; Elliot Foster. Sub (used): Krisztian Gagyi.
Referee: Kieran Graham.