Spencer Weir-Daley's first minute goal had looked like creating a rout, but bottom side Hinckley were resolute and eventually equalised through Theo Brown after the break.
But further goals from Ian Ross, Marc Newsham and Weir-Daley again rendered Richard Conway's late strike irrelevant.
United had carved out some excellent opportunities at the Greene King Stadium but were thwarted a greater tally of goals following some wayward finishing and good goalkeeping from home debutant Ashley Timms.
Dennis Greene had opted to make five changes ahead of kick-off, with three changes in the back four being supplemented by Anton Foster's return from suspension and injury alongside debutant Greg Mills.
United led inside 53 seconds. Newsham whipped over a cross from the left and WEIR-DALEY was afforded a simple finish at the back post.
In a frenetic opening spell, Timms then denied Newsham, while ex-Hinckley keeper Dan Haystead had to be alert to parry Brown's dipping drive.
United continued to create. Tom Ward headed just over from a Jason Field corner, while Timms made excellent saves to deny Mills and Ben Milnes, while Mills soon waltzed through one-on-one but blazed over the bar.
Hinckley mustered a response with Roland Agbor firing a fraction wide from 20 yards.
Timms made two further stops before the end of the first half, with successive saves to deny Weir-Daley further rewards.
The hosts levelled on 57 minutes. Jack Lane's corner was only partially cleared and BROWN crashed home a 20-yard drive, via a deflection off Nathan Stainfield, to give Haystead no chance.
United stepped up a gear and retook the lead two minutes later. Milnes whizzed a corner straight to the edge of the box and ROSS controlled the ball before dispatching it low past Timms.
Boston sealed their win eight minutes later when a glorious move saw Mills freed down the right and his low cross was stabbed home at the near post by NEWSHAM.
The United landslide continued with a fourth goal in the 73rd minute. Substitute Charley Sanders flicked the ball into the path of WEIR-DALEY who won his personal dual with Timms to round the keeper and fire the ball into an empty net.
Mills crashed a drive over the bar, while Newsham fired wide from a Field cross, but Hinckley closed in the ascendancy.
Haystead had already saved expertly to keep out Gavin Hurren's low drive, but when Brown swung in the flag-kick, CONWAY forced the ball home from close-range to add some respectability to the scoreline for the hosts.
Hinckley (4-4-2): Ashley Timms; David Kalonji, Gavin Hurren, Mario Lopez, Mark Branch; Roland Agbor (Godwin Moraka 77), Harvey Headley, Jack Lane (Flavien Chambaud 70), Theo Brown; Luke Richards (Richard Conway 62), Jarred Pickering. Subs (not used): James Hartland, Martin Conway (gk).
Boston (4-3-3): Dan Haystead; Conor Marshall, Tom Ward (Paul Lister 78), Nathan Stainfield, Jason Field (Gary Silk 78); Ian Ross, Anton Foster, Ben Milnes (Charley Sanders 68); Greg Mills, Spencer Weir-Daley, Marc Newsham. Subs (not used): Phil Watt, James Reed.
Referee: Guy Stretton