The National League high-flyers proved to be too strong at the Racecourse Ground and took a stranglehold on proceedings with Chris Holroyd's first minute effort.
Mike Fondop-Talom and Paul Rutherford wrapped up the Red Dragons' progression before half-time.
United boss Craig Elliott had named an unchanged XI for the fourth match in succession but the Pilgrims found themselves a goal down inside 58 seconds when Nicky Deverdics pulled the ball back from the right for HOLROYD to finish from close-range.
The hosts took a two-goal lead nine minutes later when FONDOP-TALOM found space 20 yards from goal to send a low drive past George Willis.
United found their mojo around the 20-minute mark, with Max Wright lashing an effort too high from 20 yards and Ryan Qualter’s header flying narrowly wide off Mark Carrington.
The Pilgrims’ best chance fell to Gavin Allott, who headed marginally wide following Ben Davies’ cross from the right.
Wrexham broke with purpose from a Boston corner and effectively killed the contest with RUTHERFORD finishing expertly on the run.
The second half was played out in bitter temperatures and torrential rain and ex-Boston keeper Christian Dibble and his opposite number Willis made late respective saves to deny Jay Rollins and Fondop-Talom as Wrexham eased through to the second round draw.
Wrexham (4-2-3-1): Christian Dibble; Mark Carrington, Jake Lawlor, Bradley Walker, James Jennings; Akil Wright, Luke Summerfield; Nicky Deverdics, Chris Holroyd, Paul Rutherford (Matt Sargent 86); Mike Fondop-Talom. Subs (not used): Stuart Beavon, Jack Thorn, Bobby Grant, Rob Lainton (gk).
Boston (4-3-3): George Willis; Ben Davies, Ryan Qualter, Spencer Harris, Ashley Jackson; Brad Abbott, Andi Thanoj, Liam Agnew; Max Wright (Jay Rollins 79), Gavin Allott (Jordan Slew 73), Nicky Walker (Lewis Gibbens 57). Subs (not used): Jonathan Wafula, Sebastian Malkowski (gk).
Referee: Simon Mather.
Attendance: 1,083 (80 visitors).