Substitute Marriott spectacularly struck in the 78th minute after Alvechurch defender David Solademi had failed to clear his lines.
The match was hardly a classic, played in hight winds and rain courtesy of Storm Darragh, but extended United's unbeaten run to five on home turf.
Graham Coughlan made five changes to his side's previous outing in the National League Cup - but the first half was completely derailed by the stormy conditions.
Ammar Dyer had Alvechurch's best opportunities before the break, crashing one effort against the crossbar and a second attempt narrowly wide.
The visitors had a further chance after the break, with Aidan Stone expertly denying Matthew Richards with his legs three minutes into the half.
Jacob Hazel was denied by Ronnie Hollingshead, before United's breakthrough finally arrived 12 minutes from time with MARRIOTT producing a moment of magic to send the Pilgrims safely through.
Boston (4-1-2-1-2): Aidan Stone; Jai Rowe, Tom Leak, Connor Teale, Sam Reed; Jordan Richards; Dylan Hill, Frankie Maguire (Sam Osborne 72); Tony Weston (Keaton Ward 61); Pemi Aderoju (Lennon Wheeldon 68), Jacob Hazel (Adam Marriott 72). Subs (not used): Kieran Coates, Michael Bostwick, Jake Lovelace (gk).
Alvechurch (4-1-4-1): Ronnie Hollingshead; Archie Crowther, Krystian Pearce, David Solademi, Ethan Sephton (Jaelon West 84); Sam Perry; Callum Sullivan (Bradley Stretton 70), Joe Willis, Matthew Richards, Ammar Dyer; Tyrone Barnett (Shay Willock 61). Subs (not used): Dieter Schwarm, Korey Peets-Fearon, Dale Eve (gk).
Referee: Lee Hible.
Attendance: 836 (22 visitors).