Fasth shot a 6-under 65 in the third round to move to 10-under par 203. His only victory to date has been at the 2000 Madeira Island Open. He is best remembered, however, for his second-place finish behind David Duval at last year's British Open.
McDowell, playing only his fourth European Tour event, shot a 66 to put him in the leading group. McDowell, who celebrated his 23rd birthday Tuesday, used a putter given to him by Darren Clarke to roll in four birdies and an eagle before a final hole bogey cost him the outright lead.
"Yesterday was pretty difficult for me, I was very protective because I was leading the tournament," said McDowell, a member of the Great Britain and Ireland team that retained the Walker Cup for the first time last year.
"Today I was much more relaxed and had fun out there. It helped being in the third last group and it will help not being in the last group as well on Sunday.
"If I think about trying to earn enough to win my card I will be in big trouble. I'm going to give it a rip and try and make some birdies and give the others something to think about."
McDowell won nine college tournaments while attending the University of Alabama.
Sluman is playing his first European Tour event other than the British Open and was persuaded to play after flying over from the United States to attend the wedding of Jesper Parnevik's sister Jill to Per-Ulrik Johansson.
"Jesper's been trying to get me to come over for the last few years but the wedding clinched it," said Sluman, whose wife Linda is half Swedish.
"I had my European Tour card in 1984, but unfortunately never used it because I got my U.S. card as well. It's been a wonderful experience and I hope I can finish it off with a win."
Johansson also enjoyed a successful day, carding a course record 63 to finish four shots off the lead.
Celebrity Families of 2014 [PHOTOS]