The Black Caps, five times World Cup semi-finalists without ever reaching the final itself, last won a major international one-day tournament when triumphing in the 2000 Champions Trophy, 'the mini World Cup', in Kenya.
And Vettori believes it's high time a side ranked as the world's third-best 50-over team lifted another big prize, starting with the 2008 Champions Trophy which is due to take place in Pakistan in September.
"We are a good one-day side, we've got most areas covered," said Vettori. "Now it's about time for us to win one of these tournaments.
"Australia are still the team to beat but with the retirements of Glenn McGrath and Brad Hogg, who was a very good one-day bowler for them, and Adam Gilchrist, maybe they've come back a little bit."
The Black Caps, in only their second one-dayer at the 'home of cricket' beat England by 51 runs to complete an impressive revival after being thrashed by a massive 114 runs in the opening contest of this five-match series.
That loss at the Riverside was a heavy blow for a team featuring several inexperienced players to absorb, coming as it did on the back of a 2-0 Test series defeat against England and a reverse by nine wickets in the one-off Twenty20 international.
Then came the controversy of their no-result washout at Edgbaston, the game needing just one more over to be a 'live' match when the umpires, with Vettori's men on the brink of levelling at 1-1, called it off because of rain.
But despite being bowled out for just 182 at Bristol, New Zealand did get on equal terms with a 22-run win.
And they managed to just do enough at The Oval to get over their anger following the controversial run-out of Grant Elliott, who made an immediate impact following his late call-up from English club side Weybridge, after the all-rounder was knocked to the ground by Ryan Sidebottom, to win by one wicket off the last ball.
"From Edgbaston onwards we've played some really good cricket, and to get a result 3-1, as convincingly as that, overseas means you have a pretty happy dressing room," said Vettori.
"It was good to see half a dozen of our senior guys have stood up at one time or another and been the leading performer."
Scott Styris led the way on Saturday with a dashing 87 off 91 balls and put on a run-a-ball 77 with fellow all-rounder Jacob Oram as New Zealand made 266 for five after losing the toss.
Then teenage quick Tim Southee, named man of the series for taking 13 wickets at an average of under 19 apiece, again demonstrated his knack of getting rid of key batsmen by dismissing Kevin Pietersen for six on his way to a return of three for 49.
"Scott Styris and Jacob Oram took it away from England a little bit," said left-arm spinner Vettori, who weighed in with three for 32.
He added: "Kevin Pietersen is obviously their best batsman and when you get a marquee player like him in the opposition side, everyone lifts."
"We know Tim Southee's got the talent and he will have learned a lot."
The elated 19-year-old, who thanked seasoned team-mate Kyle Mills for acting as his "mentor" on the field, was delighted by his award.
"It's an awesome achievement and something I'll remember forever. I just tried to sit in, not go for too many runs and build pressure that way."