Elton Chigumbura's side, however, also showed that they would be a dangerous opponent for many sides with opener Chamu Chibhabha (64), Hamilton Masakadza (80) and Brendan Taylor (40) showing they were all highly competent international batsmen.
They had been on 214 for three and well set to push on for the final 15 overs before Taylor's dismissal effectively ended their resistance and they were bowled out for 277 in 48.2 overs.
"I'm happy with the result but credit must go to Elton and his boys, they really put us under pressure," South Africa captain AB de Villiers said.
"They also batted very well and probably just lost their way midway through their innings."
Zimbabwe had earlier exploited the slow nature of the pitch to put the World Cup favourites under pressure, but in reality they lost the game in the final 10 overs of South Africa's innings when they conceded a staggering 146 runs.
The 25-year-old Miller moved from 50 to 100 in 28 balls and after he had brought up his second one-day international century he attacked Solomon Mire in the 48th over, smashing 30 runs.
He finished on 138 and the nine sixes in his 92-ball knock were also a World Cup record.
"We really went for too much in the end," Zimbabwe captain Elton Chigumbura said. "We didn't get the wickets we needed in the middle period and paid for it in the end.
"We were in the game for a long time so it was a good sign for the games to come."
South Africa's vaunted pace bowlers Dale Steyn and Morne Morkel failed to really trouble Zimbabwe's batsmen when they began their innings.
Steyn did not even open the bowling and his first five overs cost 38 runs for no wickets as he bowled too short while the wicket nullified a lot of his pace. He finished with 1-64 from nine overs.
Leg-spinner Imran Tahir was a constant threat and he took the wickets of both Chibhabha and Masakadza before he finished with figures of 3-36 from 10 overs.
(Reporting by Greg Stutchbury in Wellington; Editing by Patrick Johnston/Sudipto Ganguly)
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