The pair also discussed North Korea and Russia's invasion of Ukraine at the talks in Bali, a day before the G20 summit on the Indonesian island.
Both said they opposed the use of nuclear weapons in Ukraine. Mr Xi, who has been urged to talk down Vladimir Putin, reiterated China's calls for peace while adding there was "no simple solution to a complex problem".
Mr Biden said in his press conference "it's difficult to say that I'm certain that China could control North Korea". But he also told Mr Xi that China had "an obligation" to dissuade Pyongyang from engaging in another nuclear weapons test.
Taiwan was a major focus during their three-hour meeting at a luxury hotel shortly after Mr Xi's arrival.
Claimed by Beijing, the self-governed island counts the US as an ally, and has always been a thorny issue in US-China relations.
Tensions spiked in August when US House Speaker Nancy Pelosi visited Taiwan. China responded with large-scale military exercises around the island, prompting fears of a possible conflict between the US and China.
A readout to Chinese state media on Monday said Mr Xi had stressed that Taiwan remained "the core of China's core interests… and the first red line in US-China relations that cannot be crossed".
In recent weeks US officials have warned that China may escalate plans to invade Taiwan.
Reporters on Monday asked Mr Biden if he believed this to be true, and if he thought a new Cold War was brewing.
"I absolutely believe there need not be a new Cold War. I have met many times with Xi Jinping and we were candid and clear with one another across the board. I do not think there is any imminent attempt on the part of China to invade Taiwan," he said.
"I made it clear we want to see cross-strait issues to be peacefully resolved and so it never has to come to that. And I'm convinced that he understood what I was saying, I understood what he was saying."Mr Biden said the two leaders had agreed to set up a mechanism where there would be dialogues at key levels of government to resolve issues. Secretary of State Antony Blinken will also be visiting China soon, he said.
He added that he had made it clear to Mr Xi that "our policy on Taiwan has not changed at all. It's the same exact position that we have had".
Mr Biden has repeatedly said the US will defend Taiwan if it is attacked by China. It has been seen as a departure from the long-held US policy of "strategic ambiguity" over Taiwan, under which it does not commit to defending the island. Officials have rowed back on his statements.
Mr Xi, in China's readout, called on the US to "be consistent with its words and actions" over Taiwan.
The US has long walked a tightrope over the Taiwan issue. A cornerstone of its relationship with Beijing is the One China policy, where Washington acknowledges only one Chinese government - in Beijing - and has no formal ties with Taiwan.
But it also maintains close relations with Taiwan and sells arms to it under the Taiwan Relations Act, which states that the US must provide the island with the means to defend itself.