Leading scorer Murray took his side 13 points clear at the top of League One by expertly dispatching a low shot across keeper David Lucas in the 64th minute to earn the Seagulls their 10th home win in a row and score his 19th goal of the season.
Irish midfielder Gary Dicker put Brighton ahead with a 12th minute penalty after Craig Noone was brought down by Andy Frampton.
But spirited Swindon levelled midway through the first half thanks to a wonder strike by David Prutton, who scored with a tremendous 25-yard drive into the middle of the goal.
Brighton were awarded their second penalty after 61 minutes when Frampton was adjudged to have pushed Murray but Dicker's poorly-struck spot-kick was saved by Lucas.
Swindon were unlucky moments later when a fierce shot by Mike Grella cannoned back off the underside of the bar. But Murray restored the Seagulls' lead by superbly drilling home across Lucas and into the far corner.