The Newcastle United attacker opened the scoring from the penalty spot in the 16th minute and added further goals after 30 and 40 minutes.
Frankie Maguire scored a fantastic goal for the Pilgrims two minutes after half-time, but United were unable to muster a comeback.
United boss Graham Coughlan had made nine changes ahead of kick-off, with Connor Teale and Alex Lankshear the only survivors from Saturday's 3-1 win at Gateshead.
Newcastle took the lead after Kieren Donnelly brought down marauding full-back Charlie McArthur in the box and NEAVE stepped up to confidently beat Bobby Jones, despite the United keeper guessing correctly.
A flurry of Boston chances arrived in response. Donnelly's angled drive was charged behind by Cathal Heffernan, visiting keeper James Taylor denied Michael Kelly and Liam Waldock in quick succession, while Jordy Hiwula crashed an effort too high from around the penalty spot.
The Pilgrims cheaply surrendered possession on the half hour mark with NEAVE advancing to beat an exposed Jones with aplomb.
NEAVE completed his treble 10 minutes later, with another highly composed finish - and then sailed an audacious, angled lob on to the roof of Jones' net moments before half-time.
Boston cut the deficit two minutes after half-time, no doubt with Coughlan's words still ringing in their ears, with MAGUIRE chopping infield to beat Taylor with a fine, curling finish from around 20 yards.
Chances thereafter were scarce, with the Magpies comfortably holding on to record their second win in the competition so far.
Former Swindon Town, Forest Green Rovers and Yeovil Town full-back Marcel Lavinier, registered on Monday to play in the fixture, made his United debut as a 62nd minute substitute.
Boston (4-3-3): Bobby Jones; Ben Grist, Connor Teale, Alex Lankshear (Marcel Lavinier 62), Michael Kelly; Frankie Maguire (Billy Bradley 75), Rocco Coyle, Liam Waldock (Dylan Hill 62); Kieren Donnelly, Jordy Hiwula, Manni Norkett (Owen Hampson 62). Subs (not used): Luca Alonzi, Joe Batty, Rhys Lovett (gk).
Newcastle (4-2-3-1): James Taylor; Ciaran Thompson, Miodrag Pivas (Joe Brayson 78), Cathal Heffernan, Charlie McArthur; Scott Bailey, Anthony Munda; Alfie Harrison, Sean Neave (Sam Alabi 86), Seungsoo Park (Dylan Charlton '70); Kacey Wooster. Subs (not used): Logan Watts, Fraser Harper, Mohammed Waddani, Adam Harrison (gk).
Referee: Emily Heaslip.
Attendance: 630.
PIC: Chris Bray.