New Zealand faced a difficult chase on a sharply turning pitch and were 5-96 on Monday (Tuesday AEST) before Tom Latham made 37 in an unbroken partnership with Mark Bracewell (24no) to guide the Black Caps to their target with 104 balls remaining.
New Zealand's 400th win in ODIs levelled the five-match series at 1-1 after West Indies won the first match, also at the Guyana National Stadium, by seven wickets.
Lennox claimed his first five-wicket haul in seven one-day internationals after a belated international debut at 31 and his figures from eight overs were the third-best by a New Zealand spinner in ODIs.
He set in motion a collapse which saw West Indies lose all 10 wickets for 65 runs after a solid opening stand between John Campbell (43) and Akeem August (18).
Lennox shared with Mitchell Santner the key wickets of Keacy Carty who made 95 and captain Shai Hope who made an unbeaten 87 in a partnership of 131-runs in the first match on Saturday.
Santner bowled Carty for 18 with a ball which pitched on middle and turned sharply to beat the bat, while Lennox bowled Hope for seven with a ball that pitched on middle and leg and straightened past the inside edge.
Amir Jangoo's 24 was the only other major contribution for West Indies who lost their last five wickets in 10 overs.
Henry Nicholls (17) and Will Young (28) gave New Zealand a brisk start, putting on 35 for the first wicket at almost five runs per over.
In a double setback, the Kiwis lost both Young and Mark Chapman (0) in quick succession. Chapman was run out second ball, taking a chancy run from a push to cover only to be beaten by Vitel Lawes' quick and accurate throw.
Nicholls was out with the total 3-52, before Daryl Mitchell and Latham combined to put on 42 for the fourth wicket. Mitchell made 28 and seemed likely to be in at the close but charged a ball from Khary Pierre and was stumped.
Dean Foxcroft (0) was bowled by Pierre (2-33) shortly afterwards, raising the pressure on the tourists, but Latham and Bracewell, who passed 1000 runs in ODIs, held out until the end.
