New Zealand led by three at the end of the first quarter but a tactical change by Neville, who shifted Harten and Helen Housby up front and introduced Natalie Haythornthwaite, made a dramatic difference as England stormed back to lead 24-21 at the interval.
The match was the sixth world cup final in a row between Australia and New Zealand over the past 20 years.
The Proteas reached the final at the 1995 Netball World Championship in Birmingham where they won the silver medal after losing to Australia in the final.
They will either play England again, New Zealand or Australia in the semifinals. The Proteas have emerged as one of the on form sides in the tournament, winning five consecutive matches to march on to tomorrow's semi-finals.
"It's massive for the silver ferns, for Netball NZ and just for the general community back home".
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"As a team, we are very proud of what we have achieved in this tournament, I think we have done very well".
It was a bitter disappointment for England coach Tracey Neville, whose brother Phil recently suffered a similar fate as coach of the England's women soccer team that lost in the World Cup semi-final to the United States.
New Zealand stayed in control in the key moments and was cool when the pressure came on. She admitted. "We seemed to be chasing the game which is something we haven't done in this tournament".
"Every day was going to be a challenge for us".
"Just to see the growth of players from where they started in August to where they are now - even us fossils - it's been unreal".
Team England were also victorious in their third-place play-off game, beating South Africa 58-42.