The 20-year-old right-arm paceman claimed 6-16, the best figures for any bowler on his ODI debut, to destroy Bangladesh, who were 160 all out in the 37th over after a rain-delay had reduced the contest to 40 overs a side.
South Africa romped home with nearly nine overs to spare, bringing Bangladesh down to earth after the hosts' recent ODI series victories at home to Pakistan and India.
For the visitors, Faf du Plessis top-scored with an unbeaten 63 as he and Rilee Rossoew (45 not out) took the team home after Rabada's scintillating display with the ball.
Armed with pace and accuracy, Rabada conceded two runs in his first over before returning to wreck the hosts who were left to rue their decision to bat first.
He uprooted opener Tamim Iqbal's off-stump with the fourth delivery of that over before getting Litton Das caught at midwicket and trapping Mahmudullah lbw to complete his hat-trick.
"I did not know what to bowl (when on a hat-trick)," Rabada said after collecting the man-of-the-match award in his fairytale debut.
"Maybe a bouncer, normal length or yorker and I missed it by a mile. It was a bit of fluke but I will take it."
Bangladesh left-arm spinner Taijul Islam was the first player to claim a hat-trick on his ODI debut at the same Shere Bangla National Stadium late last year against Zimbabwe.
Shakib Al Hasan (48) and Mushfiqur Rahim (24) tried to rebuild the hosts' innings and Nasir Hossain contributed 31 down the order but Bangladesh could not last their quota of overs.
Rabada went on to claim the wickets of Soumya Sarkar, Mashrafe Mortaza and Jubair Hossain to register the best ODI figures by a South African bowler.
The second match is scheduled for the same venue on Sunday.
(Reporting by Amlan Chakraborty in New Delhi; Editing by Ken Ferris)
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