The flanker went over at the end of both halves in a 24-12 win for the visitors as the Kings' hopes of climbing off the foot of the table evaporated in their last home match of the season.
Their ill-tempered triumph, punctuated by a series of scuffles between the teams, came after the Stormers had seen their slim hopes of a playoff spot disappear just before kick-off when the Cheetahs beat the Auckland Blues in Bloemfontein to move out of reach.
But despite the blow their injury-hit squad still turned in a motivated performance with Fourie's pair of tries proving the difference.
His first came two minutes from the end of the first half in a pushover and the second in the 78th minute as he broke from the scrum and drove through several tackles for a fine individual effort.
Fly-half Elton Jantjies added 11 points from the boot with another penalty converted by first choice kicker Joe Pietersen, whose game time was limited by a cut on his lip that required stitches.
Kings' fly-half Dimitri Catrakilis, who moved from the Stormers last season, kicked all his team's points.
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