Slimbridge came away from their first visit to Sixways Stadium with three points after Harrison Reeves' 87th minute penalty saw them nick the game at the death.
Slimbridge had gone ahead through George Neill's strike in the 20th minute, but the hosts hit back through Zaqib Hussain on the half hour.
Slimbridge restored their lead three minutes later with Neill notching his second, and the lead lasted until the hour mark, when Daniele Reka scored Raiders's second equaliser.
With three minutes remaining, Neill was felled in the box from a clumsy tackle by Charlie Foley and Reeves stepped up to convert the penalty and claim the win for the Swans.
A full report and reaction from Supporters Club Man of the Match Sam Blackie to follow.