Forster fastened onto a pass from Adam El-Abd to fire the Seagulls into a third minute lead.
Swindon goalkeeper Petr Brezovan twice denied the impressive Dean Hammond before the equally influential Jake Robinson increased Brighton's lead just before half-time.
Bas Savage laid back a searching cross from George O'Callaghan and Robinson, who had a fine match in midfield, buried his 20-yard shot.
Swindon taunted by loud shouts of "You don't know what you're doing" from the stands, were disorganised and disjointed. They continued to hoof long balls up front to lone striker Blair Sturrock, all to no avail.
Swindon's horrid afternoon was completed 17 minutes from the end when Hammond fired home Brighton's third.
He breezed past two defenders before guiding a fine solo effort past Brezovan to push the Seagulls a step nearer the play-off places.
Brighton have lost only once in the last 12 games, whereas Swindon are facing an uncertain Christmas and New Year as the proposed takeover of the Wiltshire club by the Andrew Fitton Consortium seems no closer to completion.
Irate Swindon fans protested long and loud after the match calling for hasty changes in the boardroom.