News of the death came as authorities said they have freed hundreds of people arrested when riot police dispersed the protesters with volleys of tear gas and water cannons on Saturday.
The opposition Democratic Action Party (DAP) identified the dead man as Baharuddin Ahmad, who it said "passed away... from breathing difficulties during the rally."
It was unclear if his death was related to the use of tear gas on the protesters, the DAP said, adding its secretary general Lim Guan Eng will visit Baharuddin's family.
Police spokesman Ramli Mohamed Yoosuf confirmed the death but said the man was a bystander who had died of a heart attack.
"The death has nothing to do with the demonstration. He died due to heart attack. There was no external or internal injuries," Ramli told reporters.
Ramli also said that all 1,667 people arrested during the protest, including legislators and rally leaders, were freed around midnight Saturday after demonstrators had dispersed.
Among those freed were Ambiga Sreenivasan and Maria Chin Abdullah, top leaders of Bersih, the broad coalition group that led Saturday's rally to demand electoral reforms ahead of elections expected next year.
Abdul Hadi Awang, president of the Pan-Malaysia Islamic Party (PAS), the country's largest opposition grouping, and Nurul Iman -- the daughter of opposition leader Anwar Ibrahim -- have also been released.
Anwar, who was not arrested but was injured when he was knocked down during the chaos, has been released from hospital after being kept in overnight for head and leg injuries, his aides said.
Normality returned to Kuala Lumpur late Saturday after police dismantled barriers put up around the city in a massive security lockdown ahead of the protest, the biggest in Malaysia since 2007.
Organisers said 50,000 people joined the protest, while police estimated there were 10,000 demonstrators in total.