Rizwan's half-century followed, off 32 balls and they went on to become the first opening pair to post three century stands in T20 World Cup history before Azam fell for 53 off 42 balls.
Pakistan's seven-wicket win capped a remarkable comeback from the 2009 T20 World Cup winners who had been on the brink of elimination after last-ball defeats to India and Zimbabwe in their first two matches.
"The guys have worked hard and we have always believed," said Rizwan.
Pakistan beat the Netherlands and South Africa, but still needed the Proteas to choke against the Dutch for an unlikely lifeline, which they capitalised on by beating Bangladesh in their final Super 12 game.
'Go after the new ball'
"I don't think we have seen our best yet, which is probably the scarier part for whoever faces us," said former Australia opening batsman Matthew Hayden, who has been a mentor to the team.
"Sky's the limit. You can never beat down class."
Pakistan's bowlers set the tone after Azam lost the toss. Aided by some razor sharp fielding they kept New Zealand to a below par 152-4.
Rizwan became the second wicket to fall with 21 still needed, but Mohammad Haris's 30 off 26 balls helped complete a deserved, if nervy, victory with five balls to spare.
"Obviously, me and Babar decided to go after the new ball and the pitch was difficult," said Rizwan, who was named player of the match.
"When we finished the powerplay, the discussion was for one of the guys to go deep."
New Zealand, who topped Group 1, were aiming to make their second consecutive final, but their hunt for a maiden World Cup again fell short.
Teams batting first had won five from six games played in Sydney this tournament and when New Zealand captain Kane Williamson won the toss he had no hesitation in asking Pakistan to bowl, but it proved to be tough going.
In an eventful opening over, Finn Allen hit Shaheen Afridi for four off the first delivery then was given out the next ball lbw.
'Tough pill to swallow'
It was overturned on review because for Afridi to promptly do the same again and this time it was plumb.
Conway was run out for 21 by Shadab Khan and they suffered a big blow when Glenn Phillips was caught and bowled by Mohammad Nawaz for six.
Daryl Mitchell and Williamson took more risks after reaching the 10-over mark at 59-3, with the first six of the innings coming in the 13th over.
Williamson fell on 46 to an Afridi yorker before Mitchell, unbeaten on 53, and Jimmy Neesham, who scored 16, added 29 runs in the final three overs.
"At the halfway mark we thought we had a competitive total but it was disappointing not to make Pakistan work harder," said Williamson, who saw his usually sharp fielding side shell a series of catchable chances, including Azam when on nought.
"It's a tough pill to swallow," he said.
"I think if we want to be honest, we wanted to be more disciplined with our areas. Like I said, Pakistan deserved to win."