Lindani Ndwandwe of South Africa and Britain's Chris Lloyd were in a tie for third place on 61, one ahead of a group of 10 players.
The opening event of the 2012-13 European Tour season made a late start after play was washed out on Thursday and Friday because of the waterlogged state of the Royal Durban Golf Club.
Organisers decided on Saturday to cut the tournament from 72 to 36 holes and to reduce the course to a par-65. Clark, teeing off on the 10th hole at 0720, was quickly out of the blocks.
The 36-year-old South African notched a hat-trick of birdies from the 13th before mixing three more birdies with a solitary bogey coming home.
"I wasn't sure what to expect, whether having more par- threes would make it easier or not," Clark said. "It wasn't easy out there but I was amazed at how good the greens were with all the rain we've had.
"They were pretty firm and great to putt on. I made some good putts and that's really what gave me my score."
Clark, who finished second at the 2006 Masters and has won three times on the European Tour, led for most of the round until Madsen made his move towards the end of the day.
The Danish rookie, who graduated from Tour School last week, carded three birdies in his last five holes to tie for the lead.
"I was really excited to get going. This is my dream and I couldn't wait to get started," said Madsen. "I like my chances now.
"It's a good position that I'm in. I just have to go out tomorrow and focus on my own game and play it shot by shot."
The Durban course sits nearly 15 metres below sea level and has suffered unseasonably high amounts of rain in recent months. The €1 million event is co-sanctioned by the Sunshine Tour.