Xi and Biden shook hands in front of the two nations' flags before starting a long-awaited sit down on the Indonesian resort of Bali ahead of a Group of 20 summit.
Biden said that Beijing and Washington "share responsibility" to show the world that they can "manage our differences, prevent competition from becoming conflict."
Xi told Biden that the world has "come to a crossroads."
"The world expects that China and the United States will properly handle the relationship," Xi told him.
"As the leaders of our two nations," Biden said, " we share responsibility, in my view, to show that China and the United States can manage our differences, prevent competition from ... turning into conflict, and to find ways to work together on urgent global issues that require our mutual cooperation."
Responding to Biden, Xi said the relationship between their two countries was not meeting global expectations.
"So we need to chart the right course for the China-U.S. relationship. We need to find the right direction for the bilateral relationship going forward and elevate the relationship," Xi said.
"The world expects that China and the United States will properly handle the relationship," he said, adding he looked forward to working with Biden to bring the relationship back on the right track.
Both nations are increasingly suspicious of each other, with the United States fearing that China has stepped up a timeline for seizing Taiwan.
US officials said ahead of the meeting that Biden hoped to set up "guardrails" in the relationship with China and to assess how to avoid "red lines" that could push the world's two largest economies into conflict.
The most sensitive issue is Taiwan, the self-governing democracy claimed by China.
Biden has said the meeting should establish each country's "red lines", and the overarching goal will be setting "guardrails" and "clear rules of the road", a senior White House official told reporters hours before the summit.
"We do all of that to ensure that competition does not veer into conflict."
US President Joe Biden told his counterpart Xi Jinping that China's "aggressive" actions on Taiwan put peace at risk, during a landmark meeting in Bali, Indonesia on Monday.
After a three-hour summit, the White House said Biden had raised objections to China's "coercive and increasingly aggressive actions toward Taiwan" adding they "undermine peace and stability across the Taiwan Strait and in the broader region."
The Biden - Xi meeting comes ahead of the G20 summit, which opens on Tuesday in Bali, where both leaders and the heads of 18 other economically powerful nations meet to discuss global issues and policy.
Xi and Biden have spoken by videoconference five times since the US leader took office but the Chinese president's last in-person US summit was with Donald Trump in 2019.
Beijing wants Washington to "work together with China", foreign ministry spokeswoman Mao Ning said Monday.
She called for the United States to "appropriately keep differences in check, promote mutually beneficial cooperation and avoid misunderstandings and mis-judgements in order to push US-China relations back on track for healthy and stable development."
Russian President Vladimir Putin is staying away from the G20 meeting. Instead, veteran foreign minister Sergei Lavrov will represent Moscow.
Russian Foreign Minister Sergei Lavrov on Monday dismissed an news agency report that he had been taken to hospital with a heart condition, scolding Western journalists for what he cast as false reporting.
Officially, neither the war in Ukraine, nor Putin's dark threats to use nuclear weapons are on the summit agenda, but the conflict will dominate discussions.
Russia will also be under pressure to extend a deal allowing Ukrainian grain shipments through the Black Sea when the agreement expires on November 19.
British Prime Minister Rishi Sunak will urge the agreement be renewed and call for "a G20-wide commitment never to weaponize food production and distribution", Downing Street said.
At a minimum, Biden and his allies would also like to see the G20 make it clear to Putin that nuclear war is unacceptable.
But even a clear statement on this issue from the grouping is likely to be blocked by a mixture of Russian opposition and Chinese unwillingness to break ranks with its ally in Moscow or give Washington a win.
Ukrainian President Volodymyr Zelenskyy - invited as a compromise with host Indonesia - will address the summit by videoconference, a day after a triumphant visit to Kherson, a key city taken back from Russian forces.
The G20 has always been more comfortable discussing finance and economics than security and Moscow would like it to stay that way.
"We categorically reject the politicization of the G20," the Russian foreign ministry said Sunday, offering a taste of what leaders might hear from the famously unbending Lavrov.
After the West imposed the most stringent sanctions in modern history on Russia for attacking Ukraine, Lavrov said Moscow would turn away from the United States and its allies and instead expand relations with countries in Asia, Africa and Latin America.
Soaring energy and food prices have hit richer and poorer G20 members alike -- and both are directly fueled by Putin's war, as are African nations.
AU chief Macky Sall, also the president of Senegal, plans to attend the G20 summit to voice Africa's concerns to the gathering of economic powers.
G20 ministerial meetings leading to the summit have failed to agree a final joint communique and Indonesian officials said Monday it remained a "work in progress" and a "main goal" for the summit.
AU chief Macky Sall, also the president of Senegal, plans to attend the G20 summit to voice Africa's concerns to the gathering of economic powers.
"Honestly, I think the global situation has never been this complex," Indonesian government minister Luhut Binsar Pandjaitan said Sunday.
"If eventually (the G20) leaders do not produce a communique, that's that, it's OK."
This report was compiled with data from Reuters and Agence France-Presse.