Slagter entered the final stage with a 13-second cushion over Spaniard Javier Moreno and all he needed was to stay out of trouble.
The Dutch 23-year-old rookie did exactly that over the 90 kilometre city circuit to finish 17 seconds ahead of Moreno.
Olympic team pursuit gold medallist Geraint Thomas, who surrendered the overall lead to Slagter on Saturday and slipped to the fifth place, was third, 25 seconds back.
Lotto-Beliso rider Greipel staved off Slagter's Blanco team mate Mark Renshaw's challenge to claim a record 14th Tour Down Under stage victory that came a week after his triumph at People's Choice Classic.
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