WASHINGTON—Congress approved a rewrite of rules touching every corner of finance, from ATM cards to Wall Street traders, in the biggest expansion of government power over banking and markets since the Depression.

The bill, to be signed into law soon by President Barack Obama, marks a potential sea change for the financial-services industry. Financial titans such as J.P. Morgan Chase & Co., Goldman Sachs Group Inc. and Bank of America Corp. may be forced to make changes in most parts of their business, from debit cards to the ability to invest in hedge funds.

The Senate passed the bill 60-39 Thursday, following House passage last month. Earlier in the day, three northeastern Republicans joined with Democrats to block a filibuster, allowing the bill to squeak through.

Now, the legislation hands off to 10 regulatory agencies the discretion to write hundreds of new rules governing finance. Rather than the bill itself, it will be this process—accompanied by a lobbying blitz from banks—that will determine the precise contours of this new landscape, how strict the new regulations will be and whether they succeed in their purpose. The decisions will be made by officials from new agencies, obscure agencies and, in some cases, agencies like the Federal Reserve that faced criticism in the run-up to the crisis.