Instead, Governor-General Jerry Mateparae will represent New Zealand in Singapore.
"It was a difficult decision because obviously I want to go to both," Key told reporters.
He described Lee as "a close and long-time friend of New Zealand" and said that Mateparae, as New Zealand's head of state, was more senior than the prime minister.
Key said he had decided before New Zealand's semi-final, when they beat South Africa with only one ball to spare, that if his nation made the final he would attend.
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"Our two countries have a special rivalry. Over the years we have been treated to some truly exceptional sporting clashes and I expect this one will be no different."
However, his Australian counterpart Tony Abbott is bypassing the cricket and will join other regional leaders in Singapore for Lee's funeral.
Lee "was a leading international statesman of his time, and laid the foundation upon which successive Singaporean and Australian governments have built a strong relationship," Abbott said in a statement last week.