Elliott, who turned 36 at the weekend, hit a six off the penultimate ball to take New Zealand into Sunday's final in Melbourne where they will face either Australia or defending champions India.
"It is great. I don't think this win is for myself, or the team, it is for everyone here. The support has been amazing," said Elliott, who smashed Dale Steyn into the stands for the match-clinching six.
"When you have 40,000 fans screaming at you every ball, it has been an absolute pleasure playing at Eden Park and playing in front of the home crowd.
"We have had a good run, this is the first final we have been in and we will approach it as any other match."
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Skipper Brendon McCullum made a 26-ball 59 to also help the Black Caps to their rain-adjusted target of 298 after South Africa had made 281 for five batting first in a match reduced to 43 overs per side.
But the rain came at the worst possible time as they were well-set at 216 for three in the 38th over when play was halted for two hours.
"These boys, I am so proud of them. A lesser team would have laid down today. To see Grant come in and be as calm as he was... We've given ourselves a chance of the big prize," said McCullum.
"Credit to South Africa, they gave as good as they got. I feel for them at the moment. But our boys were outstanding. And it was a great semi-final."
"It's really painful but we have no regrets. We left it all out on the field. We had opportunities and it will take us a long time to get over this," he said.