Clarke, who made 259 not out, declared with a first innings lead of 115 over South Africa's 450, but his bowlers were unable to claim enough wickets to force a result in a test which lost a whole day to rain.
South Africa suffered some early wobbles in their reply, but reached tea on 114 for three and pressed on to 166 for five in a quiet evening session before stumps were drawn.
Clarke was in imperious form once again as Australia padded their lead in the morning.
The Australia captain matched Cook's tally with a triple when he drove Vernon Philander through cover and then surpassed him in the next over, hitting Morne Morkel straight back down the ground for four.
Morkel did not take kindly to that, delivering a blow to his stomach with the next ball, but it did nothing to knock Clarke out of his stride.
Mike Hussey added some drama to the proceedings too as he scored his 16th Test century.
He was on 99 when South Africa launched a huge appeal for lbw, with Hussey only saved when television replays showed the slightest inside edge on Philander's delivery.
The century was then achieved despite a near run-out as he went for a quick single.
And his dismissal came from the very next ball he faced as he hit a shot straight to substitute Faf du Plessis at cover to go for precisely 100.
Matthew Wade came in to add an unbeaten 19 before Australia declared on 565 for five.
The move paid early dividends as Alviro Petersen went for just five, but Australia could not add to the pressure despite a couple of close calls before the interval.
First Graeme Smith edged a Peter Siddle delivery just short of Hussey in the slips, and then James Pattinson bowled Hashim Amla with a no-ball, the batsman called back by the umpire after he had started to walk off.
Australia continued to probe for breakthroughs in the afternoon and their eagerness to get on with the task saw them use up their remaining reviews in attempts to get rid of Amla and Jacques Kallis, the latter claim never standing a chance.
But between the two, Smith edged into the slips for 23, and Australia got the wicket they most wanted when Amla was caught by Hussey off Siddle for 38 just before tea.
Kallis reached the second interval on 40 but added only a further nine runs before edging spinner Nathan Lyon to Clarke at slip.
AB de Villiers and Jacques Rudolph batted through most of a low-key finale, until the latter was given out lbw to Lyon (two for 41) and failed to overturn the decision on review.
De Villiers finished 29 not out with Philander on one, while Pattinson took two for 58.
"It was unfortunate not to get a result today," said Clarke, who was named Man of the Match for his record innings.
"But I think we fought hard with both bat and ball after the first day."
Opposite number Smith added: "Credit to Michael and the way his team batted, they put us under a little bit of pressure on the last day but we held on.
"I think the challenge for us was always going to be the first Test, to get into the series, and we've done that and we can go on to Adelaide with confidence. We know as a team we're capable of playing better than this."
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