The Pilgrims were too strong for Evo-Stik Northern Premier Division opponents Workington, although the visitors did create a grandstand finale through Jake Simpson's spectacular late free-kick.
Joel Dixon subsequently made a terrific late save to deny Scott Allison and prevent a mammoth midweek replay trip for the Pilgrims.
The victory was soured somewhat by a hamstring injury sustained by top scorer Southwell deep into stoppage time.
Nevertheless, it was 'job done' for Dennis Greene and his side against the plucky Cumbrian underdogs.
Southwell was first to threaten when he saw his fierce near post drive beaten behind by Jonny Jamieson.
Zak Mills' 20-yard drive then deflected wide while Dan Wordsworth was forced to clear Southwell's drive off the line following Greg Tempest's inviting cross from the left.
United seized the advantage in the 20th minute when STEER thumped home a stunning left-footed effort, high to Jamieson's right, following a brilliant mazy run into the Workington box.
The United onslaught continued and Kaine Felix's fierce drive was parried behind by Jamieson.
Workington threatened meaningfully for the first time with Joel Dixon called upon to tip Gareth Arnison's header over the bar following Matty Tymon's cross.
Workington then came incredibly close to an equaliser, with Tymon's effort from the centre circle curling inches wide at the last second.
At the end of the half, Jamieson made an excellent stop to block Carl Piergianni's downward header on the line.
United made a rocky start to the second half, but gradually found their feet and Jamieson was called upon to deny Southwell and Tempest in quick succession.
The Pilgrims' second goal arrived a minute shy of the hour mark when SOUTHWELL picked himself to dispatch a coolly-taken penalty after being wiped out in the box by Jamieson.
United had chances to kill the tie, with Mills' back post drive tipped behind by Jamieson before Mills headed wide from Tempest's corner.
The Pilgrims were unable to find a clinching third and a nervy finish was ensured when SIMPSON whipped home a stunning, angled free-kick into the far top corner from out on the left wing.
Joel Dixon then made a superb save to claw Allison's header behind as the Reds threatened to take the tie to a replay.
United's cause was not aided by Southwell's late injury, but the Pilgrims held on to book a place in Monday's first round draw.
Boston (4-4-2): Joel Dixon; Liam Marrs, Zak Mills, Carl Piergianni (Michael Hollingsworth 64), Rene Steer; Kaine Felix, Scott Garner, Kyle Dixon, Greg Tempest (Jamie McGhee 83); Dayle Southwell, Mark Jones (Marc Newsham 74). Subs (not used): David Fallah, Sam Vince (gk).
Workington (4-4-2): Jonny Jamieson; Jake Simpson, Josh Calvert, Dan Wordsworth, Gari Rowntree; Conor Tinnion (Joe Mwasile 67), Phil McLuckie, Gavin Skelton, Scott Allison; Matty Tymon, Gareth Arnison (Alex Newby 81). Subs (not used): Adam Telfer, Anthony Wright, Lee Andrews.
Referee: Michael George.
Attendance: 708 (13 visitors).
PIC: Tim Hoff.