Ben Grist broke the deadlock two minutes after half-time, substitute Kieren Donnelly doubled the lead with a majestic lob in the 62nd minute and Alex Lankshear added United's third three minutes later.
Donnelly wrapped up victory in the fourth minute of additional time with another fine solo effort to make it three goals in two competitive outings to date.
Graham Coughlan had opted to make five changes ahead of kick-off, with Rhys Lovett and new signing Jordy Hiwula making their competitive debuts for the Pilgrims.
The home side started brightly, but Leeds gradually got themselves a foothold in proceedings. The Pilgrims threatened first with Hiwula crossing for Lenell John-Lewis to test Rory Mahady with a firm downward header.
Hiwula was denied a debut goal by robust defending from Reuben Lopata-White, who subsequently saw team-mate Coban Bird whistle an effort narrowly wide with the last kick of the first half.
United edged ahead after the break when Greg Sloggett's free-kick delivery was poked home from close-range by centre-half GRIST.
Sloggett almost turned provider again on the hour mark, but Jordan Richards planted a free header wide from a well-delivered corner kick.
The Pilgrims did double their advantage two minutes later with Dylan Hill winning the ball back to send DONNELLY into the clear and he did the rest with an audacious lob to give Mahady no chance.
Sub Norkett teed up LANKSHEAR to angle home United's third, off the underside of the crossbar, three minutes further on as the home side threatened to cut loose.
Mahady saves kept out Donnelly and Norkett, while Rhys Chadwick fired high and wide for the visitors.
The Pilgrims added their fourth goal with the last kick of the game with another expert, angled finish from man-of-the-moment DONNELLY.
Boston (3-4-1-2): Rhys Lovett; Ben Grist, Connor Teale, Alex Lankshear (Billy Bradley 79); Jordan Cropper, Jordan Richards (Rocco Coyle 75), Greg Sloggett, Dylan Hill; Arjan Raikhy (Zak Mills 46); Lenell John-Lewis (Kieren Donnelly 46), Jordy Hiwula (Manni Norkett 63). Subs (not used): Finley Ging, Bobby Jones (gk).
Leeds (4-2-3-1): Rory Mahady; Will Firth (Louie Dudley 74), Reuben Lopata-White (Jonny Howson 66), Louis Enahoro-Marcus (Jayden Lienou 81), Coban Bird; Alfie Cresswell, Oliver Pickles; Rhys Chadwick, Jacob Render, Devon Brockie; Harry Gray. Subs (not used): Darryl Ombang, Logan White, Lewis Pirie.
Referee: Alan Dale.
Attendance: 830.
PIC: Chris Bray.