Vettori was also responsible for two run outs while Chris Harris became the first New Zealander to take 200 wickets and score 4,000 runs in one-dayers as West Indies were dismissed for 159 in 41.2 overs.
Captain Stephen Fleming had earlier lashed 67 from 66 balls to help New Zealand to 266 all out.
West Indies made a slow start to their chase, scoring 54 for two in their first 15 overs compared with New Zealand's 83 without loss.
Their hopes suffered further when Vettori trapped Brian Lara lbw for a cautious 30. The spinner also dismissed Dwayne Bravo for four, Scott Styris taking a juggling close catch on the leg side.
Vettori had earlier ran out Devon Smith (44) with a direct hit from some distance. Harris also reacted quickly at point to run out Ramnaresh Sarwan for 19, hurling the ball to bowler Styris who nipped off the bails.
Vettori and Harris, who had Ridley Jacobs (1) caught on the boundary by Chris Cairns for his 200th wicket, then wrapped up the victory with 8.4 overs to spare in the late evening gloom.
The players left the field three times during the day as showers and bad light threatened to spill the game over into Sunday.
LOOSE CUT
New Zealand, playing in their first one-day match at Lord's, were bowled out in 49.2 overs. They had looked set for over 300 before losing their last seven wickets for 49 runs.
Openers Fleming and Nathan Astle shared a 120-run stand with Fleming striking nine fours before playing a loose cut straight to Chris Gayle off the bowling of the lively Dwayne Bravo.
Astle (57) played more conservatively and was out in the same manner, clearly annoyed with his choice of shot.
Sarwan, with one-day best figures of three for 31, and Dwayne Smith, one for 27, helped limit New Zealand after Tino Best in particular had been costly in the first 15 overs.
West Indies were continually poor in the field and Shivnarine Chanderpaul even collided with Ian Bradshaw in the deep when taking a catch off Best to dismiss Craig McMillan for a steady 52.
New Zealand became the first team in the series to win batting first after being sent in by Lara.
They won all four of their completed games in the series and will return to England in September for the ICC Champions Trophy in confident mood.
Scorecard:
New Zealand
S.Fleming c Gayle b Bravo 67
N.Astle c Gayle b Bravo 57
H.Marshall c Sarwan b Gayle 44
S.Styris c Powell b Dwayne Smith 1
C.McMillan c Chanderpaul b Best 52
C.Cairns st Jacobs b Sarwan 5
J.Oram c Jacobs b Best 15
C.Harris c and b Sarwan 1
D.Vettori c Dwayne Smith b Sarwan 6
G.Hopkins run out 0
I.Butler not out 0
Extras (lb-8 w-9 nb-1) 18
Total (all out, 49.2 overs) 266
Fall of wickets: 1-120, 2-143, 3-146, 4-217, 5-233, 6-249, 7-252, 8-265, 9-266
Bowling: Best 7.2-0-57-2 (nb-1 w-2), Bradshaw 6-1-28-0 (w-1), Dwayne Smith 10-1-27-1, Bravo 10-0-67-2 (w-3), Gayle 10-0-48-1, Sarwan 6-0-31-3 (w-3)
West Indies
C.Gayle c Styris b Oram 4
D.Smith run out 44
R.Sarwan run out 19
B.Lara lbw b Vettori 30
D.Bravo c Styris b Vettori 4
S.Chanderpaul c McMillan b Vettori 31
R.Powell c Marshall b Vettori 18
D.Smith lbw b Vettori 2
R.Jacobs c Cairns b Harris 1
I.Bradshaw run out 0
T.Best not out 1
Extras (lb-2, nb-3) 5
Total (all out, 41.2 overs) 159
Fall of wickets: 1-5, 2-45, 3-98, 4-105, 5-106, 6-144, 7-149, 8-150, 9-150
Bowling: Oram 8-2-26-1 (nb-1), Butler 6-0-25-0 (nb-2), Styris 5-0-22-0, Harris 10-0-45-1, Vettori 9.2-1-30-5, McMillan 3-0-9-0
Result: New Zealand win by 107 runs
