He added: "We are immensely proud of you - your calmness under pressure, flawless sailing, brilliant teamwork, tactical mastery and innovative genius has returned yachting's prize to New Zealand.
"Congratulations from all of New Zealand."
English and Labour leader Andrew Little gave special praise to Burling, who they said had continued the tradition of sailing greats like Sir Peter Blake.
Little also said Team NZ chief Grant Dalton had "held the team together" between regattas and backer Sir Stephen Tindall had ensured the team had the resources for a new challenge. In a reference to mountaineer Sir Edmund Hillary, Little said Dalton would be pleased to have "knocked that bastard off".
The Labour leader also fitted in a joke at the expense of Disabilities Minister Nicky Wagner, who was criticised earlier this month for saying she would rather be out on Auckland Harbour than in disability meetings.
"This genuinely was a day when we would all rather be on the harbour."
Green MP Eugenie Sage said Team NZ had "well and truly erased" the "heartache and despair" of San Francisco in 2013.
"I confess I am not a great sports fan, but the chance to watch these boats fly, powered by the wind, water and human muscle has certainly got me up early."
Sage took the opportunity to lobby for women to be included in the next event in New Zealand, saying the rules and boat designs should be changed to allow women to compete.
New Zealand First leader Winston Peters also diverted from the script to make mention of New Zealand women's sporting success in the last week, congratulating the women's rugby sevens team on their World Series victory.
Peters said Team New Zealand's success had been a triumph of athleticism, science, and technology. It was a victory for the "smarter, faster, and yet delightfully laid-back Kiwi", he said.
The campaign was especially significant because of its "Cinderella beginning" in 2013, when the Government and public's support was wavering after the loss in San Francisco.
Act Party leader David Seymour said the victory today was all the sweeter after the bitter loss to "traitorous former New Zealanders" in 2003 and the San Francisco defeat in 2013, which he blamed on then-Economic Development Minister Steven Joyce, who "went out on the water and led to us losing time after time".
Maori Party co-leader Marama Fox traced Team NZ's winning catamaran back to the double-hulled waka of 19th-century Maori in New Zealand.
"I'm going to hold back from giving a bill to Team NZ for intellectual property because we like innovation, we like to share, and we like to see New Zealanders have success on the world stage."
