Farman denied Zak Mills and Jamie McGhee to hand City the upper hand in the Bluefin Sport Lincolnshire Senior Cup quarter-final, following an initial 0-0 draw.
Mark Jones netted for the Pilgrims, but successful conversions from Bradley Wood, Craig Stanley, Matt Rhead and Shaun Harrad saw the Red Imps through.
Boston (4-4-2): Henrich Ravas; Zak Mills, Scott Garner, Carl Piergianni, Liam Marrs; Kaine Felix (Jay Rollins 82), Kyle Dixon, Grant Roberts (Jamie McGhee 70), Jordan Deacey; Dayle Southwell (Jack Friend 65), Mark Jones. Subs (not used): Jake Sheridan, Sam Vince (gk).
Referee: Rob Atkin.
Attendance: 877 (184 visitors).