The Black Caps were in trouble at 49 for five and 75 for six before being bowled out for 182 thanks mainly to all-rounder Grant Elliott's 56 in only his second one-day international and man-of-the-match Kyle Mills's career-best 47.
That still left them with precious few runs.
But a fine return of four for 38 in 10 overs from teenage quick Tim Southee, which saw the 19-year-old rip the heart of England's middle-order with three for none in eight balls, led to victory by 22 runs with 22 balls to spare here at Nevil Road.
"Our performance was one of our better ones," said Vettori whose side are now all square at 1-1 in this five-match series ahead of next week's final two matches at The Oval and Lord's.
"It has been a tough tour and this gives us a chance to win the series. If we had lost here and been 2-0 down, it would have been tough to get everyone up for the last two games in London," the left-arm spinner said.
Vettori, frustrated after Wednesday's second ODI at Edgbaston was rained-off just one over shy of the cut-off point for a result with New Zealand on the verge of a win, added: "We stole a victory from 79 for six.
"To defend a total of 180, is one of those games you really remember. It keeps us in the series.
"It is a really good feeling in the dressing room," added Vettori who saw his side lose the Test series 2-0 before being thrashed by nine wickets in the lone Twenty20 and 114 runs in the first one-dayer.
"Having your backs against the wall has been a trait of New Zealand cricket teams for a long, long time. To deliver today (Saturday), particularly with some young guys doing so well, is amazing.
"Kyle Mills had a great all-round game, Tim Southee's spell was superb and Grant Elliott has just been amazing in his first two games."
England were on top after fast bowler Stuart Broad led the attack with two for 14 in 10 overs.
But Mills's 40-ball innings, featuring two sixes and five fours, proved vital for World Cup semi-finalists New Zealand, who added 50 runs in their last five overs.
Although Southee's treble reduced the hosts to 64 for six, England remained on course until the pace bowler returned to have captain Paul Collingwood lbw for an innings top-score of 34.
"We had the opportunity to be pretty ruthless and go 2-0 up and we kept losing wickets," said Collingwood.
"Kyle Mills played really well and struck the ball well but you can't get away from the fact we were delighted with them getting 182 on that wicket.
He added: "We played some great cricket recently in the one-day form of the game and today was disappointing because we could have nailed it.
"But we didn't take that opportunity and that's something we're going to have to learn as a unit."