Chasing an unlikely 482 for victory, South Africa were bowled out for just 200 as left-arm quick Mitchell Johnson (5-59 off 16 overs) starred again.
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Johnson finished with match figures of 12-127 but Clarke was quick to spread the praise as Australia won a Test on the road for the first time since April 2012.
Shaun Marsh and David Warner both hit centuries during the match while debutant Alex Doolan produced an excellent 89 in the second innings as Australia took a 1-0 lead in the three-Test series.
"Mitchell is at the top of his game, he's fit and strong and his execution is perfect," Clarke said.
"Credit to the bowlers and batters, Alex Doolan played an amazing innings on debut."
Clarke made special mention of Australia's fielding display, with their excellent effort highlighted by Doolan's two close catches on the final day to dismiss Graeme Smith (4) and JP Duminy (10).
"Our fielding has been outstanding over the last six months," he said.
"We have high expectations and it's nice to see some results."
The victory is Australia's sixth in a row in the longest form of the game, and keeping that winning habit is important for Clarke.
"Winning is always the best start," he added. "We knew how tough a team South Africa were and it doesn't feel like an easy win when you are out there.
"You have to be on from ball one and we were.
"It's about enjoying tonight and celebrating - it's not every day you beat the (world) number one side - but we have to ready for Port Elizabeth."
The second Test begins in Port Elizabeth on Thursday.