Hudson most recently led the All-Whites to an intercontinental playoff for the World Cup, falling 2-0 to Peru over the two-legged tie.
Goal previously reported the Rapids were closing in on hiring Hudson, though what would have happened if New Zealand had won that playoff and made the World Cup was unclear.
"It’s an exciting moment and a wonderful opportunity," Hudson said in a news release. "From the start, I was impressed by the club’s culture, front office and vision for the future. I believe we share a philosophy, and now it’s just a matter of going out and getting the results that both this club and community deserve."
Hudson takes over following the August firing of Pablo Mastroeni, who led the Rapids to a surprise playoff appearance in 2016, only to suffer through a disappointing 10th-place finish in the Western Conference this season. Steve Cooke finished out the campaign as interim coach.
"This is a very special moment for the Rapids," Colorado sporting director Padraig Smith said. "In Anthony, we believe we’ve hired one of the most promising young talents and we’re extremely grateful for the support shown by the Kroenke family and the KSE executive team who remain committed to our goal of bringing another MLS Cup to Colorado."
Hudson went 9-11-7 over his three-year stint with New Zealand, leading the team to the top spot in Oceania qualifying.
He was the youngest manager at the Confederations Cup this past summer after qualifying with New Zealand through victory in the OFC Nations Cup. The team lost all three of its games in Russia.
Hudson coached the Real Maryland Monarchs in the American third tier from 2008 to 2010. He later spent time as a reserves coach at Tottenham under Harry Redknapp and had a short stint with Welsh side Newport County, which at the time was in England’s fifth tier.
He also coached Bahrain’s senior and Under-23 sides from 2012-14 before moving on to New Zealand.
Hudson becomes the eighth full-time coach in Rapids history and 10th overall.