It is their first encounter since an acrimonious phone call in February.
"We get along great. We have a fantastic relationship, I love Australia, I always have," the AFP news agency quoted Mr Trump as saying.
Protesters also gathered in Mr Trump's home city, the first time he has returned there since his inauguration.
Mr Trump said the two leaders had "reaffirmed ties" between their two countries, and discussed economic, trade, and national security co-operation, as well as migration issues.
They met on the USS Intrepid, a WW2-era aircraft carrier that was involved in the war against Japan in the Pacific.
Relations between the two leaders were strained over President Trump's reluctance to honour a deal to accept up to 1,200 refugees from Australian detention camps.
Mr Trump had reportedly said his February call on the issue was his "worst" that day, and had previously tweeted that the refugee resettlement agreement was a "dumb deal".
But the US later said it would honour the agreement made by the Obama administration.
Strained bonhomie - Nick Bryant, BBC News, New York
Travelling through New York in a limousine is nothing new to Donald Trump. But never before has he done so with the presidential seal on the side of the car. He had not set foot in New York for 107 days, the longest absence from his native city at any stage of his life.
Inevitably, in a city where fewer than one in five residents voted for him, there was a hostile reception from protesters. By Big Apple standards, however, the demonstrations were small-scale.
Nor were there any fireworks on the Hudson during his delayed meeting with Mr Turnbull, another businessman turned politician.
In fact, both men seemed so determined to shrug off their famed testy phone conversation - Mr Trump with a nonchalant air, Mr Turnbull with a rictus grin - it ended up looking like strained and faux bonhomie.
As he met Mr Turnbull, Mr Trump said the refugee bill "has been worked out for a long time", and said media coverage of the phone call was "a little bit of fake news, as the expression goes."
"We got a little testy, but that's OK," he later added.
It is thought the two leaders may have discussed North Korea as part of their talks on national security.
Tensions have been rising around the Korean peninsula, with repeated threats from North Korea and the presence of a group of US warships and a submarine.
Observers also say Australia is also concerned about how the Trump administration's America-first trade policy might affect economic relations.
In one of his first actions after taking office, Mr Trump pulled his country out of the Trans-Pacific Partnership trade deal, a 12-nation trade agreement which would have included the US, Australia and much of the Asia-Pacific.
He also said he would visit Australia, which he called "one of the most beautiful places on earth", but did not set a date.
Australia is one of the United States' staunchest allies and troops from the two countries have fought together in all recent major conflicts including Iraq and Afghanistan.
Speaking at the event on board the USS Intrepid, Mr Turnbull said the "stability and prosperity of our region over so many decades has been secured - and is secured today - by the United States", a commitment he said was renewed by President Trump.