New Zealand leads the three-match series 2-1, with Jamieson having also bowled the last over in Thursday's second match to help his side to a three-run victory.
The West Indies won the first match by seven runs, with all three matches decided in the final over.
New Zealand managed only 9-177 batting first on Sunday but seemed headed for a comfortable win when the West Indies slumped to 8-88 in the 13th over at Saxton Oval.
Romario Shepherd scored 49 from 34 balls and Shamar Springer 39 from 20 as the West Indies rallied to reach the last over needing 14 runs to win with one wicket in hand, with Shepherd and Akeal Hosein at the crease.
The onus again fell on tall quick Jamieson to bowl the final over and he held his nerve, conceding only two runs from the first four balls before dismissing Shepherd with the fifth ball to end the match.
Once again, the West Indies' tail came to the rescue of its brittle top order.
The tourists lost two wickets to Jacob Duffy in their second over, were 2-47 at the end of the first power play, then lost 6-35 through the middle overs.
Spinner Ish Sodhi took 3-34 and was player of the match.
"If you're taking it to the last over, then that means you're in the contest," West Indies captain Shai Hope said. "It's just unfortunate we didn't come out on the winning side in the last two."
Earlier, Devon Conway set up New Zealand's innings with 56 from 34 balls. The home side also suffered a late batting slump, losing 6-31 in the last five overs.
Superb fielding by the West Indies produced three run outs and Jason Holder and Matt Forde bowled excellent spells.
Forde took 2-20 and Holder 2-31 to contain New Zealand to a moderate total.
The final two games of the series take place on Monday and Thursday.
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