Hudson’s men fell at the last qualification hurdle when Peru beat them 2-0 in Lima last week to seal victory in a two-legged intercontinental play-off for a spot at next year’s tournament in Russia.
Hudson said in a statement released by New Zealand Football that he had decided not to continue with the national team after three years at the helm.
The 36-year-old Englishman said he was sad the All Whites did not qualify but praised his players’ efforts against Peru, a team ranked 112 places above them.
“The effort they have put in for this team over the years has been first class,” he said.
“I am proud of how far the team have come in the past three years and I hope that my time here has built the foundation for future success.”
Hudson, who previously coached Bahrain and Tottenham Hotspur reserves, said he was still finalising his next move.
Media reports have linked him with the Colorado Rapids in the US Major Soccer League.
Hudson successfully revamped the dour defensive style that New Zealand employed under previous coach Ricki Herbert.
But a lack of genuinely competitive matches in the weak Oceania confederation proved telling when New Zealand encountered quality opposition such as Peru.
The All Whites never lost to Oceania opposition under Hudson but could muster only one win in 16 attempts against non-Oceania teams.
New Zealand Football chief executive Andy Martin praised Hudson’s achievements and said the search for a replacement would begin immediately.
“We are disappointed to lose a top coach like Anthony, but we understand his decision and we wish him all the very best in his future endeavours,” Martin said.
“Anthony has been one of the finest coaches New Zealand Football has ever had.”
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