Hassan Khalif Shire Ali fatally stabbed one man and injured two others after fire-bombing his car in Melbourne’s Bourke Street on Friday.
Shire Ali's family said he had mental health issues for years, had refused help and was deteriorating before he was shot dead in the attack.
But the Australian prime minister said the 30-year-old was a terrorist who cannot be given excuses, as he urged imams to watch out for "infiltrators".
"It's that shady character who is at the periphery of the mosque, the one talking to young people," he told Sky News on Monday.
"These people prey on vulnerable Australians, vulnerable young men particularly.
"I think in many cases they will (know who to watch for), and what I'm saying is you can't look the other way."
Shire Ali was known to federal police and had his passport cancelled in 2015 amid concerns he planned to travel to Syria to fight with Daesh (ISIL or ISIS).
But his family said he didn't have terrorist connections.
"Please stop turning this into a political game. This isn't a guy who had any connections with terrorism but was simply crying for help," they said in a note handed to reporters.
Morrison said he wants to work with Islamic communities to ensure they are aware of when and where radicalization can happen.
"My encouragement to the Muslim religious communities across Australia is to raise the level of awareness and alertness," he said.
Home Affairs Minister Peter Dutton again defended intelligence agencies for not being aware of the lone wolf attack.
He said 14 attacks have already been thwarted, and 400 people are being watched as a high priority.
Labor spokeswoman Michelle Rowland highlighted the role all Australians can play in fighting extremism.
"Everyone in the parliament is united in ensuring that the first responsibility of us as public office holders is to keep Australians safe and ensure we do everything in our power in order to make that happen," she told Sky News.