The defining moment arrived three minutes before half-time when Joe Anderson diverted Dan Mooney's cross beyond the reach of goalkeeper Kelechi Chibueze.
It was Chibueze who proved to be the difference between the Pilgrims and a much more convincing winning margin following a string of fine saves from the Sunderland stopper.
The result moved the Pilgrims up to fourth in the early Group C standings, following their opening night 4-3 defeat at the hands of Leeds United.
Chibueze saved to deny Dylan Hill inside the first 25 seconds, with Sunderland responding through Trey Ogensuyi, who crashed an effort wide.
Frankie Maguire's rising drive flashed to safety off the angle of post and crossbar, while Chibueze made quickfire saves to deny Jordan Richards and Keaton Ward.
United edged in front when Mooney got away down the right and fizzed a ball across which was sliced into his own net by ANDERSON.
Adil Aouchiche, Sunderland's most senior player, teed up Ogensuyi for a chance to equalise after the break, but the Black Cats' centre forward lifted his effort over the top from close-range.
Sunderland's best chance fell to full-back Zak Johnson 11 minutes from time - with an effort clipping the outside of Aidan Stone's post.
Boston (4-2-3-1): Aidan Stone; Jai Rowe, Michael Bostwick (Tom Leak 63), Zak Mills, Cameron Green; Jordan Richards, Dylan Hill; Dan Mooney (Kieran Coates 63), Keaton Ward (Adam Marriott 68), Frankie Maguire; Pemi Aderoju (Jacob Hazel 63). Subs (not used): Harvey Tomlinson, Jude Frostick, Jake Lovelace (gk).
Sunderland (4-2-3-1): Kelechi Chibueze; Zak Johnson, Ben Crompton, Joe Anderson, Tom Lavery (Jenson Jones 86); Jayden Jones, Ben Middlemas; Rhys Walsh (Aaron Chung 86), Adil Aouchiche, Jack Whittaker (Finn Geragusian 86); Trey Ogensuyi. Subs (not used): Tymur Tutierov, Luke Bell, Joe Neild, Ben Metcalf (gk).
Referee: Will Davis.
Attendance: 414.
PIC: Chris Bray.