The waka made only two stops, at Tubuai and Mangareva, both in French Polynesia, and navigated using the sun, stars, moon, currents and marine life.
The sailors first sighted Rapanui at 1.34am on Thursday (Easter Island time) after more than three months at sea.
Karl Johnstone, director of the NZ Maori Arts and Crafts Institute, said it was fitting that Mr Busby would be on board as the waka arrived.
"This journey has been Hector's dream. His skills and knowledge of traditional navigation techniques are unsurpassed and it is only fitting he should be there to experience the joy of this accomplishment."
Mr Johnstone said it would be "incredibly moving" to see the waka sail into Rapanui, knowing the journey had been 20 years in the making and had helped forge cultural, scientific and academic bonds across the Pacific.
Chief navigator Jack Thatcher said spirits were high as the crews neared their destination.
"Everybody on board is buzzing and can't wait to get to Rapanui. It's a surreal feeling," he said.
The 20 sailors will rest for a week before travelling to Tahiti where the waka will lay over for the cyclone season. Most of the crew will fly home then return in April to sail back to Auckland.
Waka Tapu ("sacred canoe") was organised by the Maori Arts and Crafts Institute in Rotorua and Northland's Te Taitokerau Tarai Waka.