Andrew Strauss's hundred was the centrepiece of England's run chase as they reached their victory target of 294 for the loss of four wickets just before tea on the fourth day at Old Trafford here Monday.
But for much of this match England, now 1-0 up in the three-Test series after a draw at Lord's, were outplayed.
It needed left-arm spinner Monty Panesar's career-best haul of six for 37, which saw New Zealand dismissed for just 114 in their second innings, to give England's batsmen a chance of winning the game.
Vaughan, praising a first-wicket stand of 60 in the second innings between Strauss and fellow left-hander Alastair Cook, told reporters: "Chasing 294, the key was the start.
"If we lost early wickets we were going to be dead and buried."
Vaughan, who shared a second-wicket stand of 90 with Strauss as England completed their second highest victory chase on home soil after their 315 for four against Australia at Headingley in 2001, added: "We knew we were chasing the game.
"Monty Panesar's first three or four overs, he was nervous and didn't get his pace right. But then he got on a roll. It's amazing how a game can change."
Strauss, whose Test career had looked like coming to a close during March's tour of New Zealand, was at a loss to explain his return to form.
"It's hard to explain, that mystery thing form. The only way to get it is by scoring runs," Strauss said after his second century in three Tests.
This was Strauss's 12th Test hundred and the Middlesex left-hander said: "It is certainly one of the more satisfying.
"It's not very often you get the chance to get a hundred in the fourth innings. It's special for me, even more special for the team. We were under the pump for two days."
Strauss was dropped for England's pre-Christmas tour of Sri Lanka after a run of low scores.
Although recalled in New Zealand, the 31-year-old was still under pressure for his place come the final Test in Napier.
But, after a first innings nought, he scored a Test-best 177 and England went on to win by 121 runs as they clinced the series 2-1.
"There were a couple of technical things I worked on, not trying to hit too early through offside," Strauss said.
"There is no magic formula - you've just got to play each ball as it comes and be mentally disciplined enough to do that."
New Zealand had been 85 for two in their second innings, the third of the match, and on the brink of putting the game beyond England's reach before Panesar sparked a collapse that saw seven wickets lost for 29 runs.
"The third innings is always a tricky one in which to bat, especially when you've got a lead," said Vaughan. "You can get ahead of the game and relax a little bit. If you do, you can lose wickets in clusters."
New Zealand captain Daniel Vettori, who for the second match running took five wickets in an innings, was dejected after a match where Ross Taylor's career-best 154 not out in the first innings had put the Black Caps on top.
"When you turn up on the fourth day and have the game dragged away from you, it's very disappointing," Vettori said.
"We were 85 for two, looking very comfortable, so to be bowled out for 114 was a tough pill to swallow."