New Zealand led by 155 runs after their first innings of 575-8 declared then set West Indies the record run chase when declaring on 306-2 in their second innings.
Latham and fellow opener Devon Conway became the first openers in the history of Test cricket to score centuries in both innings as they put their side in a commanding position.
Conway, who scored 227 and 100, became the 10th batter in Test history to score a double hundred and a hundred in the same game, while Duffy took 23 wickets in the three-match series.
West Indies captain Roston Chase made just two and five in the third Test to complete a difficult series during which he scored only 42 runs and took three wickets.
“I think I had a tough series, very below par for my standards,” said Chase. “I didn’t really lead from the front on the field.
“Leading in terms of words and encouraging and inspiring the team, that was all good and well, but in terms of going out there and producing for the team, I thought I let myself down and the team down as well.”