The match revealed that New Zealand have a bright prospect in fast-medium Doug Bracewell, whose five for 85 helped to secure the win.
Vettori's three for 71 gave him match figures of eight for 141 and with 40 and 31 runs earned him the man of the match award.
The pitch at the Queen's Club didn't deteriorate as New Zealand would have hoped and it was a battle of attrition.
Zimbabwe's weak lower-order batting let them down just when it needed to play its part, but the test suggested that as a team progress is being made.
New Zealand's was an up-and-down performance.
There were good runs from Martin Guptill, a 76 and 76 double from captain Ross Taylor, 117 runs in the match from Kane Williamson, an encouraging 68 on debut from Dean Brownlie and strong bowling performances from Bracewell and Vettori.
But there were holes elsewhere. Offspinner Jeetan Patel kept getting hit over the boundary, eight times altogether, six by the aggressive Taylor, and finished with none for 142 off 36 overs.
The batting was patchy, and work needs to be done in the next four weeks before facing Australia in Brisbane on December 1.
Still, they should enjoy the win - the first since February last year against Bangladesh in Hamilton, with four losses and three draws in between - which ended a tour of five wins from six games across the three forms.
"It was a hell of a test match," Ross Taylor said. "We were in a position where we could either go for it or hold back. Full credit to Zimbabwe for putting us under pressure. I think I got a bit of luck with the bowling changes."
Vettori praised Brendan Taylor's innings "and it was only once we got him out that we believed we could win. Luckily the ball kept low in the final session but we bowled really well, especially Doug Bracewell."
Zimbabwe's coach, former Surrey batsman Alan Butcher, made a valid point for his team.
"To even contemplate a win under those circumstances shows how much we have progressed as a team," he said.
Brendan Taylor could savour a tremendous personal series, with two centuries and a 75 in the three ODIs, and a test double of 50 and 117, his second test ton in the space of three matches.
"It was a great day of test cricket, we showed a lot of character and determination," he said.
"We could have blocked out the game an hour before the end but we played positive cricket. All in all everyone can be happy with their efforts."
Ross Taylor concurred: "Going forward, Zimbabwe cricket's fortunes will change."
Just how much they have progressed will be tested when they reciprocate the trip, with a test, three ODIs and two T20s in New Zealand from late January to mid-February.