The Cherries seized the initiative from the moment Wade Elliott put them ahead in the first minute.
Elliot received a pass from Warren Cummings on the halfway line and ran on unhindered before beating Rhys Evans with a low right foot shot from 20 yards.
Swindon were visibly rocked but should have equalised a minute later when Grant Smith somehow headed over from 10 yards when it looked easier to score.
Elliot was booked in the 13th minute for a challenge on the goalkeeper and, five minutes later, Evans was forced to limp out of the action to be replaced by Steve Book.
Lee Holmes shot across the face of the Bournemouth goal, but it was a rare moment of Swindon aggression in a first half dominated by Bournemouth.
It was the same story after the break and it was no surprise when Bournemouth scored again in the 56th minute. Brian Stock's pass found a huge gap in the Swindon defence and James Hayter lobbed in his 18th goal of the season.
Bournemouth suffered a major setback ten minutes later when Cummings was carried off on a stretcher after a challenge by Steve Jenkins but even after the reshuffle Bournemouth were still on top.
In the 74th minute, Bournemouth scored their third goal and it was a sweet moment for their Swindon-born defender Matthew Mills who was unmarked when he headed in a corner taken on the right by Stock.
Swindon were doomed but they staged a late and brave rally in a desperate bid to salvage some respectability.
Smith twice had shots blocked by the legs of goalkeeper Neil Moss and, in the 80th minute, the same player hit the bar.
In injury time, substitute Stefani Miglioranzi came close to snatching a consolation, but his low shot from six yards was hacked off the line by James O'Connor to enable Bournemouth to record their biggest away win of the season.