Rusedski sealed his his victory with a love service game, although the crucial moment had come when he broke Enqvist in the sixth seed's first service game of the deciding set.
Rusedski, ranked 10 in the world and seeded four here, had overcome a mini break in the first set tie-breaker to break Enqvist twice and clinched the set on his first set point as the Swede netted a volley.
However, the 23-year-old Enqvist, who has slipped to 15th in the rankings from a high of six, put Rusedski's service under pressure immediately in the second set and obtained the vital break in the second service game as Rusedski put a volley into the net.
Enqvist clinched the set without being troubled on his serve - he had two love games during the set - on the first of his two set points.
But Rusedski dug deep and fought back in the third set, breaking Enqvist on his first service game to lead 2-0, with a brilliant backhand pass down the line which left the Swede sprawled on the court after a desperate dive to return the ball.
Rusedski had only one worrying moment, when he had to save two break points with Enqvist netting a forehand and then the Briton put away a precise volley to save the second one.
He then strolled through the remainder of the set to record his third success in four meetings with the Swede and earn a place in today's semi- finals against the Czech Republic's Petr Korda, who beat the Frenchman Lionel Roux 6-3, 6-4.
In the doubles quarter-finals, Tim Henman, Britain's No 2, and the Swiss Marc Rosset bt Sweden's Peter Nyberg and David Prinosil of Germany 7-6, 6-7, 7-5.Reuse content