The EBA called on national financial regulators to ensure that capital shortfalls would be quickly resolved.
Five Spanish banks failed, as well as one in Austria and two in Greece.
On Wednesday, Germany's Helaba pulled out of the stress tests, effectively making it the ninth bank to fail.
Eight banks named In Austria, the Oestereichische Volksbank failed the test, while in Greece two state-controlled banks - ATEbank and EFG Eurobank - fell at the hurdle.
In Spain, Catalunya Caixa, Pastor, Unnim, Caja3 and CAM failed, with seven others just scraping through the test.
Jason Karaian Economist Intelligence UnitBanks will be under pressure to build their capital buffers, regardless of how they fared in the stress test”
However, Bank of Spain governor Miguel Angel Fernandez Ordonez said there was no need to inject further capital into the banks as the sector was already undergoing a fundamental restructuring.The EBA added that 16 banks only just passed the tests. All the banks should "promptly" take steps to strengthen their financial cushion, the EBA said.
After the EBA announcement, the Bank of Portugal said two of the country's banks would immediately begin bolstering their finances.
Banco Comercial Portugues, the country's largest listed bank, and Espirito Santo Financial Group, will strengthen their balance sheets within three months.
Debt-heavy Portugal took a 78bn euro (£68bn) bail-out earlier this year. Its economy is forecast to contract 4% over the next two years.
The news came just as Italy's parliament approved a 70bn euro austerity package. The country's central bank said that all Italian banks had passed the tests with "an ample margin".
Default A key benchmark for passing the test was whether the banks have at least 5% "core tier 1" capital, which describes the best form of capital a bank can hold to make up any losses.
One analyst told the BBC that, while some people would find the results reassuring, others would see them as evidence that the tests were not credible.
He said that demands from the financial authorities that banks began immediately to bolster their core capital "is an acknowledgement that there is a risk of sovereign default".
The failed banks would have to find an estimated 2.5bn euros in new funding by the end of the year, he said.
But analyst Jason Karaian of the Economist Intelligence Unit said that the total extra funding needed by banks would be far higher in the long run.
"Given that the markets are rewarding safety and security over growth and risk, banks will be under pressure to build their capital buffers, regardless of how they fared in the stress test.
"In the end, it would not be surprising to see hundreds of billions of euros raised in the coming quarters, with the most frenzied activity centred on banks with the greatest exposure to the euro area's wobbly periphery," he said.
As expected, the four UK banks passed the test - Royal Bank of Scotland, HSBC, Barclays and Lloyds Banking Group.
The Financial Services Authority said: "The results support our own stress tests and we are pleased that the major UK banks have capital above the minimum required in the test, reflecting the work we and the banks have undertaken to improve resilience since the crisis."
Stringent The tests are a key element in fighting Europe's debt crisis, intended to identify weak banks and ensure they are made robust enough to survive a possible default on government bonds by heavily indebted countries such as Greece.
However, the tests did not consider the impact of Greece defaulting, something some analysts believe is increasingly likely.
There have been concerns, including from the ratings agency Standard & Poors, that the tests were not strict enough. However, the EBA said they were more stringent than those it carried out last year.
In 2010, both Irish banks tested, Bank of Ireland and Allied Irish Bank, were given a clean bill of health. But just months later, AIB needed a government bail-out.
Each country's national banking regulators carried out a test that simulated what would happen to a bank's finances during a recession where growth falls more than 4 percentage points below EU forecasts.
On Wednesday, German bank Helaba withdrew from the stress tests to avoid public failure.
It said it would have passed the test if regulators counted a debt-equity hybrid, called "silent participation", as a capital reserve, but the EBA, having initially said it would accept this, then changed its mind.
"Under the EBA conditions the bank failed, that is clear," Helaba spokesman Wolfgang Kuss told the AFP news agency on Friday. "From our point of view we were successful," he added.
Meanwhile, the leaders of the 17 eurozone countries will hold an emergency summit next week in a bid to agree a deal on a second bail-out for Greece, the EU president announced Friday night.
Herman Van Rompuy called the meeting after disagreement over the contribution of banks and other private investors to a second rescue package.
The disagreement has overshadowed the financial markets, prompting some of the biggest share price falls for months.