A statement from New Zealand football says the 36-year-old Englishman was out of contract after the All Whites were beaten 2-0 in Peru a week ago and has decided not to continue with the national team.
Hudson said: "I've enjoyed every second, it's been a pleasure and an honour to work with this team and to truly experience what it means to be a proud Kiwi."
"As a group we were obviously sad to not achieve our goal of qualifying for the World Cup, but from where we were three years ago to where we were in the dying few minutes of the game against Peru, I have to say I'm incredibly proud of all the players and staff, they were magnificent."
New Zealand Football chief executive Andy Martin said Hudson should be proud of the work he has done with the national team and has left his own chapter in All Whites' history.
"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," said Martin.
Martin said the search for Hudson's replacement as All Whites coach will begin immediately.
