Beauchamp and Tricks were both on target towards the end of the first half against Pinxton, while Briers applied the icing to United’s cake in the 88th minute.
The Pilgrims will now play host to Stourport Swifts in the third qualifying round.
United had the better of the early proceedings. Tricks headed wide from a Stuart Whitaker cross in the third minute, before planting another headed chance wide following Whitaker’s corner.
Pinxton’s James Green dispatched a shot too high from 30 yards, while Briers soon saw an effort blocked by keeper Daniel Hatfield following another Whitaker assist.
United took the lead on 32 minutes. Whitaker delivered a corner from the right and after the ball had been partially cleared, BEAUCHAMP lifted home an effort into the roof of the net.
Superb last-gasp challenges from Lewis Sturman and Niall Delahoy prevented Greg Marriott from netting an instant equaliser — and the Pilgrims were soon two goals to the good.
Charley Sanders supplied an excellent cross from the left and TRICKS drove the ball home at the near post.
United stepped off the gas in the second half and were lucky to see Cameron Short drill an effort off target after being afforded the freedom of the penalty area.
The second half was a scrappy affair — and United did not add their third goal until two minutes from time.
Tricks delivered a superb ball from the right and BRIERS arrived at the back post to silence the Pinxton faithful.
Jake Platten crossed for Sanders in stoppage time but Hatfield saved well, while Sanders slid another effort wide following Briers’ cross from the left.
Boston (4-4-2): Marc Hipwell; Niall Delahoy, Lewis Sturman, Jordan Nuttell, Charlie Rossington (Todd Devlin 55); Joe Briers, James Reed, Jake Beauchamp, Stuart Whitaker (Jake Platten 75); Charley Sanders, James Tricks. Subs (not used): Scott Langan, Alex Marr, Sam Towell.
Pinxton (4-5-1): Daniel Hatfield; Jack Horton, Jake Youngs, Cory Hardy, Watson Charlesworth; Lee Fazakerley (Ross Hopkinson 60), Cameron Short (Josh Graney 60), Greg Marriott, James Green (Troy Stevens 60), James Barry; Joe Wright. Subs (not used): Jordan Coupe, Scott Potter.
Referee: Martin Chester
PICS: David Shipman