Following a scoreless first half, Brian Flynn's side broke the deadlock when in-form striker Lee Trundle (pictured) gave The Swans a 1-0 lead four minutes after the break.
Four minutes later, Kevin Nugent made it two for the home side before substitute Andy Robinson sealed the three points with a third goal three minutes from the end.
The result maintains Swansea City's 100 percent record of the new campaign while The Pilgrims have just two points from their opening three league matches.
In Friday's other Third Division match, pre-season promotion favourites Northampton Town finally got off to a winning start with a 1-0 victory over struggling Darlington at the Sixfields Stadium.
Martin Smith's strike just before the break was enough to give The Cobblers all three points, as Martin Wilkinson's side moved up to mid-table after three matches.
The Quakers, meanwhile, remain rooted to the foot of Division Three after three straight defeats to begin the new campaign.
Click below for match stats:
Division Three
Northampton 1-0 Darlington
Swansea 3-0 Boston Utd
Click here for an updated Division Three table.
