First half goals from ex-Pilgrim Marc Newsham and Nathan Jarman sentenced the visitors to eventual elimination at The Northolme.
United had an abundance of chances, with Jan Budtz making superb saves to deny Dayle Southwell, Scott Garner and Nicky Walker, while Carl Piergianni and Southwell both rattled the bar inside a minute during the second half.
The Pilgrims included Kyle Dixon, Grant Roberts and Jay Rollins in three changes from the previous week's defeat to AFC Fylde. Fit-again Mark Jones was named on the bench.
United opened in lively fashion, with the pacy Rollins causing problems up front, but he was forced off through injury on 17 minutes and the Pilgrims fell behind a minute later.
Jamie Yates' long throw picked out NEWSHAM who swivelled neatly to fire beyond Jason Alexander.
The Pilgrims tried to respond and Budtz produced a brilliant save to keep out Southwell's header, while substitute Jones was heroically denied on the line by Dominic Roma.
Trinity's second goal arrived six minutes before half-time when JARMAN crashed home a brilliant 20-yard free-kick, with Alexander unsighted behind his defensive wall.
United almost cut the deficit before half-time, but Budtz clawed away Garner's goalbound drive.
Trinity defended for their lives after the break, with Piergianni and Southwell hitting the bar in quick succession from Nicky Walker corners, while Budtz denied Walker himself with an excellent low save.
Respective crosses from Roberts and sub Jordan Richards deserved much better in the closing stages but the ball would not go in for United and the Pilgrims were left to contemplate an early exit from the competition.
Gainsborough (4-4-2): Jan Budtz; Dominic Roma, Jake Picton, Adam Quinn, Josh Lacey; Liam Davis (Stephen Brogan 90), Simon Russell, Jonathan D'Laryea, Jamie Yates; Marc Newsham (Danny Forrest 77), Nathan Jarman (Nick Bignall 68). Subs (not used): Darryn Stamp, Matt Wilson, Charlie Binns, Jonathan Hedge (gk).
Boston (4-4-2): Jason Alexander (Henrich Ravas 46); Zak Mills, Scott Garner, Carl Piergianni, Liam Marrs; Nicky Walker, Kyle Dixon, Grant Roberts, Aaron Greene (Jordan Richards 72); Dayle Southwell, Jay Rollins (Mark Jones 17). Subs (not used): Jay Dowie, Harry Limb, Cameron Johnson.
Referee: Paul Graham.
Attendance: 725 (221 visitors).