Bank of England governor Mark Carney said UK banks' "substantial capital and huge liquidity" allowed them to continue to lend to businesses and households.
Chancellor George Osborne said he had briefed G7 finance ministers about the consequences of the vote.
On Twitter, he said it had been a "hard-fought campaign" and while it was not the outcome he wanted, "I will do all I can to make it work".
The referendum turnout was 71.8% - with more than 30 million people voting - the highest turnout at a UK-wide vote since 1992.
Mr Farage - who has campaigned for the past 20 years for Britain to leave the EU - told cheering supporters: "This will be a victory for ordinary people, for decent people."