The Pilgrims were 3-0 down to Basford United by the time substitute Tate, on for the injured Harrison Holland, struck at the end of the first half.
Basford added two further goals after the break to wrap up a resounding victory.
Boston (4-1-4-1): Jake Frestle (Nick Ivanov 46); Callum Rodger, Taylor Gray, Brogan Horniman (Marc Tucker 66), Nathan Stevens; Marshall Young; Jamie Gaukroger, Finlay Armond (Dougie Barrett 81), Harrison Holland (Kyle Tate 43), Jenson Bark; Will Jackson.