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Smith hit a boundary with the day's last ball to bring up his seventh one-day century, and take his side past the winning target of 179 in 36.5 overs.
Man of the match Smith needed just 118 balls for his century, with 14 boundaries and a six.
Herschelle Gibbs (57) combined in a 143-run opening stand with his captain before Shakib Al Hasan trapped him lbw to take Bangladesh's only wicket.
Hashim Amla, a Test specialist who made his one-day today, finished not out on 9.
Earlier, a half century by opener Tamim Iqbal failed to get adequate support from the rest of Bangladesh's batting order, as South Africa skittled the home side for 178.
Andre Nel claimed three wickets to take him past 100 in one-day internationals, while Charl Langeveldt, Albie Morkel and one-day newcomer Paul Harris all claimed two.
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