The Englishman, playing on a sponsor's invite in only his second DP World Tour event, fired two eagles in three holes in an eight-under-par round of 63.
That gave him a one-stroke lead ahead of Germany's Nick Bachem on a day twice disrupted by the weather as lightning threatened the area.
Bachem started at the tenth hole with a birdie four and picked up three more shots in as many holes from the 13th.
Another at the first was given straight back at the next hole but three birdies in his final five holes took him to a 64 and the early clubhouse lead.
That stood until Bradbury, having also started on the back nine, eagled the third and fifth to move to eight under.
The Yorkshireman briefly went nine under with a seventh birdie of the day at the eighth but finished with a bogey - in a round also featuring a double-bogey six at the 15th, seven birdies and seven pars.
Bradbury said: “It was pretty good to be fair – I can’t really complain with a 63 can I?
“I got a little bit of momentum going early on the back nine, then the (first) thunderstorm obviously stopped that.
“Both (eagles were) just great drives – had nine iron into them both, front flags, just hit it nice and high and it’ll stop. Then obviously you’ve got to roll the putt in.
“This is my first time really playing at altitude. Keegan, the local caddie, he’s been great getting me used to all that but yeah, the ball’s going miles. Both those par fives, I had I think 180 and 188 (yards) in - it just doesn’t make sense.”
France's Romain Langasque carded an opening six-under-par 65, with seven birdies and a single bogey at the par-three 14th, to sit third alongside compatriot Clement Berardo - who had played only 11 holes and was on a run of four straight birdies when play was suspended for the day after a second lightning stoppage.
“We know you can make a lot of birdies on this course and we also know you can make a quick few mistakes," said Langasque.
"I just avoided these mistakes, I played really well. The biggest thing I will have to do in the next three rounds is stay on my strategy and be patient, don’t make any stupid mistakes.
“It’s a new season, I love South Africa and I felt I was playing well the last two weeks. I was looking for a good round this morning and it happened so it’s cool for the confidence and I’m looking forward to the next round.”
Defending champion - and newly crowned Sir Henry Cotton Rookie of the Year - Thriston Lawrence was in a group at five under after a flawless 66 featuring three straight birdies from the fifth.
Alongside him were fellow home player Jaco van Zyl, Finland's Sami Valimaki and England's Nathan Kimsey, having earned his DP World Tour card by winning the Challenge Tour’s Road to Mallorca Rankings and then hit the opening tee shot of the new season.
A group of nine players at four under included South Africa's Jbe Kruger and Italian Francesco Laporta, who both had six holes of their first round still to play.