The call, which official Chinese news agency Xinhua said was initiated by the Labour leader, was his first publicised conversation with Xi.
Burnham took office in July after the resignation of his predecessor Keir Starmer, who oversaw the first stages of a cautious rapprochement with Beijing.
Relations had been strained for years, fuelled in part by tensions over the political situation in former British colony Hong Kong, London's removal of Chinese tech giant Huawei from 5G networks and criticism of Beijing's human rights record.
"Although the social systems and national realities in our two nations are different, our common interests outweigh our differences," Xi told Burnham, according to Xinhua.
"Both countries should show mutual respect and trust, rise above differences, seek common ground while accepting differences, and jointly expand the scope of mutually beneficial cooperation."
Burnham promised to "engage frankly and constructively on issues that matter for both countries", Downing Street said in a statement.
The British leader "raised domestic security, Russia's war in Ukraine, Hong Kong, human rights and consular cases of particular importance" during the call, it added.
The two leaders "discussed the importance of pragmatic cooperation in advancing UK and Chinese interests and addressing shared global challenges", the British readout said.
Starmer in January became the first British prime minister to visit to China in eight years, with the trip including a dozen cooperation deals and an agreement to reduce Chinese tariffs on whisky.
Britain has sought to minimise its reliance of China in recent years, notably by barring Huawei from its 5G networks and removing energy giant CGN from a nuclear power plant project.
In their Tuesday call, Xi told Burnham that "we hope that Britain will safeguard fully the legitimate rights and interests of Chinese investors", according to Xinhua.
This follows the government's nationalisation in July of British Steel, formerly owned by Chinese company Jingye Steel.
China's commerce ministry at the time opposed the move, accusing Britain of "forcibly" taking over the company "under the pretext of national security". Jingye demanded compensation and threatened the UK government with legal action.
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