The right-hander scored 61 off 48 balls as the Windies made a formidable 143 for six in their 20 overs after the Kiwis won the toss and opted to field.
Cooper's impressive knock included five fours and two sixes before she was caught by Rachel Priest off Sophie Devine, who took four wickets for 22 runs at the western Indian city's Wankhede Stadium.
“Congrats to my amazing team. We worked really hard,” Cooper said afterwards.
“This is our fourth semi-final (at a World Cup) and to finally cross this border... I hope I score another 61 in the final,” she added.
Captain Stafanie Taylor was the West Indies' second-highest scorer with 25, while Deandra Dottin chipped in with 20 before New Zealand could only make 137 for eight in their 20 overs.
Off-spinner Taylor was the star of the West Indies' bowling attack, taking 3-26.
New Zealand lost openers Suzie Bates and Priest for just 12 and six runs respectively, dealing early blows to their chances of successfully chasing 144.
Sara McGlashan top-scored with 38 while Amy Satterthwaite notched 24 runs in a 59-run fourth-wicket partnership that took the White Ferns close, but both holed out to Shemaine Campbell in the deep from successive Taylor deliveries.
The lower order could not muster the power needed to see them over the line, and New Zealand followed their men's team in going unbeaten through the group stages, but losing in the semi-finals.
The West Indies will take on three-time defending champions Australia, who edged past England in the first semi-final, in Sunday's title showdown in Kolkata. – AFP
Original source: #WT20 Windies women into World Twenty20 final