The Pilgrims dominated large swathes of their first-ever visit to Broadhurst Park and seized control through Felix's third minute opener and Southwell's 50th goal for the club five minutes before half-time.
The visitors saw numerous other chances go begging and were forced to endure an agonising finale once Sam Sheridan had reduced the arrears in the first minute of stoppage time.
Conner Robinson was back in the United starting XI ahead of kick-off, with Malachi Lavelle-Moore making way.
Boston got off to a dream start with less than three minutes on the clock when Carl Piergianni's raking ball was nodded on by Southwell and FELIX did the rest with a firm prodded finish beyond David Carnell.
Fabian Spiess made an instinctive save to beat away Sheridan's piledriver before Piergianni's effort was clawed aside by Carnell following Zak Mills' long throw as the sides traded chances.
George Thomson's effort for the hosts took a big deflection off Scott Garner and looped narrowly over before Joe Burgess inadvertently headed the ball beyond his own keeper Spiess but recovered to deny substitute Matt Wolfenden on the line.
Chances continue to arrive in an open encounter and Southwell shot wide at the back post after Piergianni had initially connected with Mills' long throw.
The visitors' second goal arrived five minutes before the break when Grant Roberts' precise throughball found SOUTHWELL who did the rest with a firm finish for his 20th goal of the season.
Boston created an abundance of chances after the break, but failed to extend their lead. Southwell's effort from outside the box deflected behind for a corner, Piergianni's looping header took a nick as it nestled on the roof of the net and Roberts' fierce, angled effort cleared Carnell's crossbar from 15 yards.
Lewis Hilliard's curling shot then flew wide following Roberts' deep cross from the right, Southwell's deflected header skimmed the bar and a further Southwell effort flew wide at the near post following Felix's cross.
FC United sub Tom Greaves tested Spiess with two routine efforts as the hosts appeared to run out of ideas, but once SHERIDAN had crashed home a spectacular volley in added time, the Pilgrims were forced to defend two further corners as nerves began to fray.
FC United (4-2-3-1): David Carnell; Dale Tonge, Tom Eckersley, Luke Ashworth, Chris Chantler; Sam Sheridan, Scott Kay; George Thomson (Greg Daniels 64), Harry Winter (Matt Wolfenden 7), Jerome Wright; Dale Johnson (Tom Greaves 73). Subs (not used): Adam Thurston, Tom Brown.
Boston (4-3-3): Fabian Spiess; Zak Mills, Nat Brown, Carl Piergianni, Joe Burgess; Scott Garner, Grant Roberts, Lewis Hilliard; Kaine Felix, Dayle Southwell, Conner Robinson (Cameron Johnson 71). Subs (not used): Ryan Robbins, Malachi Lavelle-Moore, Harry Limb, Harry Payne (gk).
Referee: Tom Bramall.
Attendance: 3,544.