Mitchell's seventh ODI century lifted New Zealand to 7-269 as they batted first after losing the toss on Sunday.
Sherfane Rutherford (55) led the West Indies' reply with his seventh half-century and an unbroken 53-run partnership between Justin Greaves (38) and Romario Shepherd (26) helped the tourists take their chase into the final over.
The West Indies finally needed nine runs from the last two balls bowled by Jacob Duffy and managed only one, finishing on 6-262.
Batters from both teams struggled in difficult conditions at Christchurch's Hagley Oval.
Timing was awkward on a two-paced pitch on which few batters were able to score freely. A strong breeze made the longest square boundaries mostly inaccessible. New Zealand also made effective use of the short ball.
The West Indies became bogged down early in their reply. After losing John Campbell (4) in the third over, Alick Athenaze and Keacy Carty consumed almost 18 overs in a partnership of 60 for the second wicket.
Athenaze made 29 from 58 balls and Carty 32 from 67.
Captain Shai Hope lifted the scoring rate with 37 from 45 balls and Rutherford's 55 from 61 added momentum before he was out in the 46th over.Â
"It was a tricky surface in terms of the extra bounce some balls had and I guess there wasn't a fluent way you could go about your one-day innings," Mitchell said.
"You had to find a way to find ways to put pressure on them at times and absorb pressure at other times."
Mitchell put on 67 for New Zealand's third wicket with Devon Conway (49) and 69 for the fifth wicket with Michael Bracewell (35) in the best of New Zealand's partnerships.
Batting at No.4, the 34-year-old Mitchell lasted until the final over, overcoming a groin injury that required on-field medical treatment when he was 79 in the 41st over.
