Southwell opened the scoring for the Pilgrims at the Bill Stokeld Stadium and eventually scored United's winner, with Jamie McGhee also on target.
Joe Brothwell and Junior Ogedi-Uzokwe scored Carlton's goals.
United edged in front on 13 minutes when SOUTHWELL met Zak Mills' right-wing cross with a well-timed header.
The Pilgrims had already suffered an injury blow by this point with Jay Dowie limping off with suspected ankle ligament damage.
Carlton levelled on 39 minutes when BROTHWELL capitalised on some poor defending to force the ball beyond Sam Vince.
United re-established their advantage just a minute later with McGHEE side-stepping two challenges inside the box to finish well but Carlton were level again before the break when OGEDI-UZOKWE nipped past Carl Piergianni to equalise.
Boston clinched the victory on 69 minutes when SOUTHWELL beat keeper Liam Mitchell at his near post with a powerful drive.
Carlton (4-4-2): Liam Mitchell; Robert Nnaji (Ryan Hastie 80), Kallum Keane, Martin Ball, Tyler Guy; Joe Brothwell (Benn Minshall 80), Alex Troke (Matt Turner 80), Josh Rae (Danny Elliott 75), Ricky Gardner (Kameron Campbell 70); Junior Ogedi-Uzokwe, Aaron Hooton (Maksym Kowal 80).
Boston (3-5-2): Sam Vince; Scott Garner, Carl Piergianni (Karlton Watson 46), Duran Reynolds; Zak Mills, Conor Marshall, Jay Dowie (Michael Hollingsworth 9), Danny Meadows, Rene Steer; Scott Garner, Jamie McGhee (Stefan Galinski 68). Sub (not used): Matt Gledhill (gk).
Referee: Declan Bourne.