The Pilgrims' progress was cemented by a first-half demolition job on opponents reduced to 10 men by the 18th minute dismissal of keeper Tom Heaton.
After that Swindon's task was virtually impossible.
Boston had the better of the opening exchanges and in the eighth minute a powerful header by Simon Rusk forced Heaton into a neat save.
But the Swindon keeper was helpless when Boston took a 10th minute lead. A low centre from the right by Brad Maylett fell to Julian Joachim whose fierce 15-yard drive was diverted high into the net off Rory Fallon.
Nine minutes later Swindon's problems increased when Joachim burst clear and was brought down by Heaton which cost the keeper a red card.
Substitute goalkeeper Matty Bulman emerged from the bench and did well to save Jason Lee's spot-kick.
However, the reprieve was only temporary as Joachim cashed in on more Swindon hesitancy as he swivelled past Bulman and hooked the ball in from 18 yards.
Swindon struggled to find a way back but almost cut the deficit when Andy Gurney tried his luck from distance only to see the ball bounce wide off an upright.
Boston replied with a third goal on 40 minutes when Maylett fired low and hard beyond Bulman following a left-wing cross by Austin McCann.
Deep into first-half stoppage time Lee headed Boston's fourth when stooping to meet Ben Futcher's knock back from a David Noble cross.
With nothing to lose Swindon started the second half with some useful attacks and Sean O'Hanlon and Christian Roberts both saw efforts flash inches off target.
Boston also went close when Noble and Chris Holland linked well down the left with the latter's cross volleyed over by Joachim.
Swindon threatened again in the 75th minute when Roberts fired a 25-yard free kick which Nathan Abbey held well.
Swindon kept pressing and Abbey saved with his legs to deny Charlie Comyn-Platt.
But he was unable to keep out an 83rd minute penalty by Fallon after Alan White had handled.