The measures, which will also include restrictions on the Russian central bank's international reserves, will be implemented in the coming days, the nations said in a joint statement that also vowed further action to come.
"We will hold Russia to account and collectively ensure that this war is a strategic failure for Putin," the leaders of the European Commission, France, Germany, Italy, Great Britain, Canada and the US wrote in the statement on Saturday.
"Even beyond the measures we are announcing today, we are prepared to take further measures to hold Russia to account for its attack on Ukraine," they added.
The move comes after the US and its allies slapped sanctions this week on major Russian banks as well as on Russian President Vladimir Putin himself as Moscow's forces pushed into the heart of Ukraine toward Kyiv.
"As Russian forces unleash their assault on Kyiv and other Ukrainian cities, we are resolved to continue imposing massive costs on Russia. Costs that will further isolate Russia from the international financial system and our economies," said Ursula von der Leyen, president of the European Commission, the EU's executive.
Cutting Russian banks out of the SWIFT system will stop them from conducting most of their financial transactions worldwide and effectively block Russian exports and imports, she said.
The sanctions will also stop Russia from "using its war chest", paralysing the assets of its central bank, freezing its transactions and making it impossible for the central bank to liquidate its assets, von der Leyen said.
It was not immediately clear which Russian banks would be removed from SWIFT, but von der Leyen said the move would ensure those selected were "disconnected from the international financial system" in a way that would "harm their ability to operate globally."
The allies also pledged to limit the sale of citizenship via so-called golden passports used by some wealthy Russians to gain residency in Western nations and access to their financial systems.
EU foreign ministers will discuss the sanctions package at a virtual meeting on Sunday evening, the fourth time they come together in a week.
In Britain, Prime Minister Boris Johnson said the allies' "decisive action" would shut Russia out of the global financial system.
"We will keep working together to ensure Putin pays the price for his aggression," he wrote on Twitter.
AAP with The Project