Prime Minister Jacinda Ardern said she had last visited Moore just two days ago.
"The world lost a man with a huge intellect, and huge heart today," she said in a statement.
Ardern said he never lost his interest and passion for politics, because he saw it as such an important vehicle for change.
Former prime minister Helen Clark, who was Moore's deputy, said he was a "passionate New Zealander" who spent his life serving New Zealand and its values.
Former Labour finance minister, and friend of Moore's, Michael Cullen told the Herald that the former prime minister needed to be remembered as the "gifted and engaging politician that he was".
"He saw himself as the last true working-class Labour leader and could have been an inspiring prime minister with just a few more votes in the 1993 election."
But Cullen said Moore's later years were marked by an "inability to let grudges go".
"This led him to isolate himself from many former friends and colleagues unnecessarily," Cullen said.
Moore suffered a stroke in 2015 when he was New Zealand's ambassador in Washington DC and had been in declining health in recent years.
He was elected to Parliament at age 23 – at the time, one of the youngest MPs in New Zealand's history.
Before becoming prime minister, Moore was New Zealand's foreign minister and later, served as Ambassador to the US.
In 1999, he was appointed the director-general of the World Trade Organisation.
Ardern said his footprint on international trade was large.
"While Mike made his mark is many ways, one of his enduring legacies to New Zealand is the work he did opening up world trade and gaining access for New Zealand exporters to new markets. That was a legacy he also left for the world".
She said Moore never lost his interest and passion for politics because he saw it as such an important vehicle for change.
Peters paid tribute to his service, saying everywhere he went, people saw the passion he had for New Zealand and connecting it to the world.
"The following quote which Mike loved was typical of his world view: 'Well it works in practice, let's see if it works in theory!'"
Former Prime Minister Jenny Shipley said she had a huge amount to do with Moore when she backed his bid for Director-General of WTO.
"I advocated strongly for him in the run up to APEC with a number of key nations. He was a shameless and compelling leader for NZ Inc," she said.
"Mike managed to transcend the line in political terms in his belief in market economies and won enthusiatic support from the left and the right. He believed in inclusive trade policy that carried everyone forward and did it his way."
Shipley said Moore was admired and loved by a wide range of people.
"His devotion to Yvonne and she to him was so clear to all who knew him in their remarkable political partnership. He will missed enormously."
