The deal struck by former president Barack Obama was publically labelled a "dumb" agreement by President Donald Trump.
Mr Cosgrove said the alliance started an "unbreakable relationship", telling Mr Pence: "We've been with you every step of the way".
Pence said the decision to go ahead with the deal was in honour of the important historic alliance between the U.S. and Australia.
US Vice President Mike Pence on Saturday reaffirmed "the strong and historic alliance between Washington and Australia" while visiting Sydney.
Under the deal signed in late 2016, the White House had agreed that Australia would send to the United States refugees from Afghanistan, Iran and Iraq who arrive at its offshore processing centres in Papua New Guinea and Nauru. The US supercarrier Carl Vinson will arrive in the Sea of Japan in days, Vice President Mike Pence said in Australia on April 22 amid high tensions with North Korea.
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I think this week is crucial for us. "If we play the way we played today then we can not be anxious ". We respect the way that we play, our philosophy. "C'mon, lets try to win the next game".
"The United States intends to honour the agreement", Pence said Saturday, but added that it "doesnt mean we admire the agreement".
Pence's visit Down Under is widely viewed as an effort to smooth over relations with Australia in the wake of a highly-publicized argument between Turnbull and Trump.
The Prime Minister has resisted those calls, and although China is Australia's most important trading partner, America remains its closest ally on security matters.
The prime minister said that "whatever the reservations of the president are", the decision "speaks volumes for the commitment, the integrity of President Trump, and your administration, sir, to honor that commitment".
Australia is one of Washington's staunchest allies and has sent troops to fight alongside the USA military in conflicts in Iraq and Afghanistan.
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Prosecutors presented more than 500 exhibits that helped tie Eric Frein (freen) to the deadly attack in Blooming Grove. After the prosecution finished its case Tuesday morning, the defense called no witnesses and presented no evidence.
Both leaders also repeatedly cited the nations' long history of military cooperation.
His statements came amid an escalation in tensions between Washington and Pyongyang due to North Korea's latest ballistic missile tests, which prompted the USA to send nuclear aircraft carrier USS Carl Vinson and its strike group to the Korean Peninsula.
Turnbull voiced his own concerns earlier this month, saying China has an obligation to use its "enormous leverage" to bring the nuclear-armed state back from the brink, toughening Canberra's position on the escalating crisis.
Australia is the last leg of Pence's 10-day Asia-Pacific tour which will conclude on Monday, when he will depart for Hawaii.
Mr Pence and Mr Turnbull also spoke of joining forces to urge China to take a greater role in pressuring North Korea to de-escalate its nuclear weapons and missile program.
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On Sunday, Pence will tour Sydney's iconic Opera House, take a boat ride along the harbor and visit a local zoo.