"When the Netherlands beat South Africa it was a significant moment for us, a very significant moment for the team in general around reaching its potential," he said on the eve of the semi-final in Sydney.
"Lots of prayers as Pakistan woke up to see that result, 232 million people can't be wrong, and as a result of that I feel there has been very much uplift in tempo in our group.
"It has been a roller-coaster... but I really believe we have yet to play our best game, which is a huge threat to oppositions."
While Pakistan's bowling, led by Shaheen Shah Afridi, has been improving, their batting has been fragile, a concern against a top-class New Zealand attack led by Tim Southee, Trent Boult and Lockie Ferguson.
Azam's form in particular remains a worry with the opener managing just 39 runs in five matches.
Hayden, himself a destructive opener who played 103 Tests and 161 ODIs, said the skipper was due some "fireworks" -- and predicted they could come on Wednesday.
"There's no question Babar has been under some adversity but that will only make him an even greater player," he said.
"We know with the weather that when there's a lull, there's often a storm that follows, so look out rest of the world because I think we're about to see something very special from Babar."
New Zealand have long been a force in white-ball cricket, but have failed to turn that into world titles, including falling at the final hurdle to Australia in 2021.
They started their campaign this year with a massive 89-run win against the defending champions, also in Sydney, and Hayden said Pakistan were wary of the Black Caps.
"They got 200 on this particular wicket against Australia... New Zealand have some really destructive players and they can put you under pressure with the bat," he said.
"They've also got a terrific, well-balanced bowling attack.
"Like New Zealand sport in general, they really punch above their weight, they believe they can win this tournament and they have the potential to do that.
"So lots of threats to our camp, no question."