Opting to field, India dismissed the visitors for 201 in 47.5 overs, and then chased down the target with almost seven overs to spare at the Bandra Kurla Complex.
Opener Raut top-scored for the hosts with an 89-ball 69, hitting 12 fours in the process.
Thirush Kamini made 43, while Harmanpreet Kaur (46) and skipper Mithali Raj remained unbeaten on 36.
Once the two openers -- Raut and Kamini -- added 119 runs in little under 27 overs, the chase was more or less sealed in India's favour. After their dismissals, Kaur and Raj played sensibly to help the hosts romp home with ease.
Earlier, Goswamy led the Indian bowlers' impressive performance with figures of three for 26, dealing an early blow to the visitors by removing opener Natalie Dodd.
The mosts important wicket of skipper Suzie Bates (27) was, however, taken by Amita Sharma.
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There were two wickets apiece for Ekta Bisht and Gouher Sultana.
Frances Mackay was the top-scorer for New Zealand with a knock of 69 off 101 balls, while Sara McGlashan made a breezy 56 off 66 deliveries. Both the batswomen hit seven fours each during their stay at the wicket.
Barring Mackay and McGlashan, none of them could contribute anything of note, which affected New Zealand badly.