Both were killed - with gunman Man Haron Monis - as security forces moved in after a 16-hour siege last week.
Australia is reviewing how Monis was not included on a watch list, despite a history of violence and instability.
Soon after the services, Prime Minister Tony Abbott warned of a heightened level of "terrorist chatter".
Speaking to reporters, Mr Abbott cautioned that the public needed to remain alert, saying that "an attack is likely", but did not raise Australia's terror alert level.
Mr Johnson and Ms Dawson will be remembered as heroes, says the BBC's Jon Donnison in Sydney.
Friends of Mr Johnson say the 34-year-old died while trying to wrestle a shotgun from the hostage-taker.
Katrina Dawson, 38, reportedly died while trying to shield a pregnant friend from gunfire.
Mr Johnson's funeral took place at St Stephen's Uniting Church, close to Martin Place, where the Lindt Cafe is based. His family and friends said they had planned a celebration of his life.
A memorial service for Ms Dawson was held at the University of Sydney.Volunteers have begun clearing the piles of bouquets that have been placed at Martin Place in tribute to the victims.
Mr Abbott has announced a review into the background and security checks on Monis.
The self-styled cleric, who was originally from Iran, was facing charges of sexual assault and being an accessory to murder.
He had a history of religiously motivated activism and displayed an Islamic flag during the cafe siege.
However, officials have said there is as yet no evidence that his actions were linked to international Islamist militant networks.
1. At 09:45 on Monday local time (22:45 GMT Sunday) police are called to the Lindt Chocolat Cafe in Sydney following reports of an armed robbery. It soon emerges a gunman is holding a number of people hostage.
2. Between 16:00-17:00, three men, then two women, sprint to safety from the cafe's side door - a fire exit.
3. Just after 02:00 on Tuesday, a loud bang is heard from the cafe and special operations officers advance towards the side door.
4. More hostages escape, running to safety on Elizabeth Street.
5. Moments later, commandos storm the cafe via a number of entrances. The remaining hostages escape.
6. Police officially confirm the end of the siege at 02:45 local time. They later report the deaths of three people, including the gunman.
