Mr Morrison also announced education savings which will see students pay a greater share of the cost of degrees.
Infrastructure projects, health and housing affordability were also high on the government's agenda.
The main opposition Labor party accused him of using a healthcare levy increase to fund tax cuts for big business.
But Labor supported the tax on the banks.
Mr Morrison said $6.2bn ($4.6bn; £3.5) would be raised over the next four years by the new levy on the big five - ANZ Bank, Westpac, National Australia Bank, Commonwealth Bank and Macquarie.
The head of the Australian Bankers' Association called the levy "a direct attack on jobs and growth".
Reset - Hywel Griffith, BBC News in Sydney
It was a textbook manoeuvre by Mr Morrison - disarm your opponents by grabbing some of their ideas.
The surprise tax on Australia's big banks and a thaw on healthcare rebates could all have come from a Labor manifesto.
Throw in a promise to spend big on infrastructure, and, according to one Australian commentator, Mr Morrison didn't just steal the opposition's clothing - he took the whole wardrobe.
In reality, this was more about steering to centre ground rather than swerving to the left.
It moves the government away from austerity measures it introduced in 2014, but remains tough on key issues like welfare payments.
Mr Morrison and PM Malcolm Turnbull will hope that is enough to keep their party room content, and ensure the government's slim majority remains intact.
"It is a tax that will hit Australians by hurting investment and could have unintended consequences... it will affect the entire banking system," said Anna Bligh, the former Queensland premier.
"This new tax is not a well thought-out policy response to a public interest issue, it is a political tax grab to cover a budget black hole."
What we knew already
The budget was the first since PM Malcolm Turnbull's government was re-elected on a tiny majority last year.
In it Mr Morrison outlined higher education reforms aimed at savings of A$3.8bn by June 2021.
But the government had already announced key aspects of the budget, including building a second Sydney airport, increasing university fees by 7.5% by 2021, and lowering the salary threshold for university debt repayments from A$55,000 to A$42,000.
In other areas, the government will provide:
The Australian Federal Police with an extra A$321m to counter the threat of terrorism;
Early childhood education with an additional A$428m;
Indigenous Australians who were exposed to radiation from British nuclear testing extra healthcare benefits;
No more subsidies on university fees for citizens from New Zealand.
Source: BBC News