The duo, who had fired the Pilgrims to victory at Aldershot Town in their opening fixture of the season, were on target in the second half to sink Wealdstone at the Jakemans Community Stadium.
John-Lewis struck three minutes after the break and Sloggett won it 11 minutes from time - after ex-Pilgrim Micah Obiero had equalised with a diving header for the visitors.
Preparing for their second match inside 48 hours, Graham Coughlan opted to hand starts to Michael Kelly and Arjan Raikhy, with Rocco Coyle and Jordy Hiwula dropping to the bench.
United struggled to get into their stride against a powerful and imposing visiting side who had won their first three matches, before losing to Carlisle United in Saturday's late kick-off.
Wealdstone twice threatened through Nathan Tshikuna early on - his first effort clipping the crossbar and the second forcing Rhys Lovett into a smart save.
United stepped up a gear after the break and half-time substitute Kieren Donnelly teed up JOHN-LEWIS to roll the Pilgrims into a 48th minute lead.
Dante Baptiste clawed aside Sloggett's 25-yard free-kick but the visitors enjoyed a spell in the ascendancy, culminating in a brilliant diving header from OBIERO to level the scoreline on 69 minutes.
Obiero rattled the post with another header two minutes later, while Lovett produced big saves to deny Omar Mussa and substitute Max Kretzschmar.
It laid the platform for United's winner - with SLOGGETT sweeping home from close-range following Hiwula's driven cross-shot from the right.
Boston (3-4-2-1): Rhys Lovett; Connor Teale, Tommy Fogarty, Alex Lankshear; Ben Grist (Rocco Coyle 85), Jordan Richards, Greg Sloggett, Michael Kelly; Dylan Hill, Arjan Raikhy (Kieren Donnelly 46); Lenell John-Lewis (Jordy Hiwula 63). Subs (not used): Jacob Scott, Manni Norkett, Billy Bradley, Bobby Jones (gk).
Wealdstone (3-1-4-2): Dante Baptiste; Jack Cook (Dom Hutchinson 61), Deon Woodman, Philip Chinedu; Alfie Massey; Enzio Boldewijn, Omar Mussa (Max Kretzschmar 78), Nathan Tshikuna, Anthony Georgiou (Sakariya Hassan 90); Sean Adarkwa (Daniel Nkrumah 61), Micah Obiero. Subs (not used): Dylan Kadji, Junior Tienisa, Lucas Covolan (gk).
Referee: Richard Eley.
Attendance: 2,077 (135 visitors).
PIC: Chris Bray.