It came by virtue of goals from Christian Roberts, Michael Pook and Blair Sturrock that eclipsed a Bradley Thomas reply for the home side.
The success kept the Robins firmly in pursuit of table-topping Walsall but left the Pilgrims anxiously looking over their shoulders at the relegation trap-door.
Boston created the first chance in the eighth minute of a game contested in blustery conditions. Francis Green fed the ball through for David Farrell to fire an angled drive a yard wide.
But it was Swindon who took the lead two minutes later when Roberts crashed the ball high into Boston's net from 12 yards after home keeper Andy Marriott could only parry aside a curling corner by Michael Timlin.
However, the Pilgrims drew level in the 17th minute when Thomas flicked the ball home after Justin Richards had fired goalwards following a free-kick from the left by home debutant Shane Nicholson.
Boston then continued to enjoy spells of pressure with Green and Richards both looking lively. But the Swindon rear-guard held firm.
Home goalie Marriott was called into action again in the 34th minute when having to stretch to clutch the ball after defensive colleague Paul Ellender had miscued a header in attempting to clear Pook's long throw.
Boston replied and twice before the break they went close - first when Drewe Broughton saw a shot blocked and then when Farrell forced a fine save from Phil Smith with a rasping left-foot drive.
The home side had to make a half-time substitution with cut eye victim Lee Canoville making way for Jamie Clarke and it was the latter who threatened in the 51st minute when bursting through only to see his 12-yard shot blocked by Ady Williams.
However, Swindon hit back and retook the lead when Jack Smith crossed from the right and Pook swept home a neat volley passed Marriott from eight yards.
The visitors pressed again in the 73rd minute and, when the ball broke to Sofiane Zaaboub, his low shot from 20 yards flashed wide of the target.
Back at the other end substitute Stephen Vaughan went close to a 79th minute leveller but fired wide with a shot from distance.
Boston exerted late pressure and won a series of corners while efforts by Green and Brad Maylett were scrambled away.
Swindon made sure of the points in the 88th minute when Sturrock - on as a substitute only seconds earlier, fired in from 16 yards after being fed by a pass from Lukas Jutkiewicz.