McIlroy, who won the U.S. Open, missed a host of putts on the front nine.
"It was getting frustrating," he said. "I was reading them and every time ... I felt as if I was hitting the putt on the line I wanted to, but it was going nowhere near the hole.
"But to birdie three of the last five holes was great, and at one stage I was behind Keegan; so to draw level with him after the first day was nice."
Bradley, who won the PGA Championship, meanwhile got off fast at Port Royal Golf Course, racing to 6 under and briefly putting Lucas Glover's course record 65 in jeopardy.
He missed putts at 14 and 16 to drop back.
Bradley played an almost flawless front nine which included eagles at the par-5 2nd and 7th, when he hit 6-iron approaches from 220-yards to within 2-feet of the pin.
McIlroy in contrast couldn't make a putt and reached the turn at 2 under following birdies at Nos. 5 and 7. His troubles on the green continued on 11, however, where he three-putted, but a 30-footer for birdie at 12 changed things.
While Bradley was starting to struggle, McIlroy found his rhythm and after driving to within 10-yards of the green on 15, was the only one of the foursome to hit the green at the par-3 16, when Clarke and Schwartzel hit their tee-shots into the sea and walked away with triple bogeys.
A heavy rain shower turned the closing two holes into a battle, but McIlroy managed to birdie 17 to draw even with Bradley and set up an intriguing final round Wednesday.
It is likely to be a two-way fight, with Schwartzel and Clarke seemingly too far back to challenge the leaders.
Clarke was typically blunt in describing his round: "It was crap and that's being polite," he said. "I didn't come here to play this badly."