The Pilgrims suffered a 2-0 defeat to the Hatters on Saturday afternoon courtesy of Danny Lloyd's brace on 35 and 90 minutes.
United's woes were compounded when defender Josh Robinson received his marching orders eight minutes from time.
County warranted their victory, which was their first in six attempts at York Street, although the opening exchanges were extremely scrappy and did not provide any pointers towards the eventual scoreline.
It took until the 28th minute for either side to craft a meaningful chance and Ross Durrant had to save well to deny Scott Duxbury, with Lloyd lashing the rebound high and wide.
County took the lead seven minutes later when Lewis Montrose freed LLOYD down the left channel with a sublime pass and the winger beat Durrant at his near post via a deflection of a sliding Robinson.
United had played without a recognised centre forward in the first half and in the absence of Gregg Smith, Murray introduced Nat Brown as an emergency striker at the break.
Brown had a header deflected wide off Montrose while Joe Fitzpatrick thumped a long-range effort past Ben Hinchliffe’s post.
Robinson received his first caution on 71 minutes and his second 11 minutes later as the Pilgrims ended with 10 men.
County sealed their victory three minutes in stoppage time when ex-Pilgrim Kaine Felix, on only two minutes from time, broke clear and squared the ball for LLOYD to tap home from close-range.
Boston (4-3-3): Ross Durrant; Kalern Thomas, Josh Robinson, Tom Batchelor, Ben Gordon; Joe Fitzpatrick, Shane Clarke, Charlie Gatter (Marcus Marshall 46); Callum Chippendale (Waide Fairhurst 61), Lewis Hilliard (Nat Brown 46), Jay Rollins. Subs (not used): Ben Clappison, Lamin Colley.
Stockport (4-2-3-1): Ben Hinchliffe; Mark Ross, Michael Clarke, Chris Smalley, Scott Duxbury; Lewis Montrose, Gary Stopforth; Sam Minihan, Jimmy Ball (Kaine Felix 88), Danny Lloyd; Kayode Odejayi (Josh Amis 60). Subs (not used): Courtney Meppen-Walters, John Marsden, Ian Ormson (gk).
Referee: Robert Massey-Ellis.
Attendance: 1,360 (228 visitors).