The Pilgrims led 2-0 at the break courtesy of Felix's 19th minute opener and Southwell's 40th minute twice-taken penalty.
Further goals were added inside a four-minute spell during the second half by fit-again Jones and debutant Hilliard.
It was a resounding victory for the Pilgrims against their play-off rivals ahead of the busy Christmas period, although they did have to withstand a strong opening spell from the visitors.
Tranmere Rovers loanee Jonny Margetts spurned the game's opening chance after being played in by strike partner Kayode Odejayi in only the second minute. There was also a crucial block from Zak Mills to deny Margetts who had wriggled through.
United's first sight of goal arrived when Scott Garner shot narrowly wide from close-range following good build-up play involving Jay Rollins and Felix.
The Pilgrims edged in front when Southwell clipped the ball through for FELIX to run on and slide across keeper Andreas Arestidou and into the far corner.
Arestidou got a crucial touch to deny Southwell on the angle before the Pilgrims were awarded a golden opportunity to double their advantage when Odejayi handled following a Grant Roberts corner.
Stockport's delaying tactics held up the taking of the penalty and despite Arestidou saving with his legs to deny SOUTHWELL, the United man made no mistake with the re-take following encroachment from the Stockport defenders.
Odejayi was unable to turn home Lewis Montrose's cross-shot after the break while Karl Ledsham's volley flew into the side-netting.
Jones' flying header was grabbed by Arestidou following Roberts' corner and United soon went three goals to the good. JONES weaved in and out of three challenges before finishing with aplomb, high to Arestidou's left, from 18 yards.
County still tried to cut the deficit with Ledsham's header parried wide by Ravas following Danny Morton's deep cross.
The Pilgrims cleared the resulting corner, with Jordan Richards doing brilliantly to nutmeg Ledsham and free Jones and Felix ran on to cross for debutant HILLIARD to sweep home three minutes into his first United appearance!
Boston (4-4-2): Henrich Ravas; Liam Marrs, Zak Mills, Carl Piergianni, Dylan McEvoy; Kaine Felix, Jordan Richards, Grant Roberts, Jay Rollins (Lewis Hilliard 70); Dayle Southwell (Mark Jones 46), Scott Garner (John Sands 77). Subs (not used): Ryan Robbins, Sam Vince (gk).
Stockport (4-1-3-2): Andreas Arestidou; Danny Morton, Sean O'Hanlon, Chris Smalley, Joe Garvin; Dale Tonge (Micah Evans 55); Wade Joyce (Abdulai Baggie 85), Lewis Montrose, Karl Ledsham; Kayode Odejayi, Jonny Margetts. Subs (not used): Gareth Roberts, Eddie Fulwood, Ian Ormson (gk).
Referee: Amy Fearn.
Attendance: 1,316 (121 visitors).