WASHINGTON — The Senate voted today to let the Federal Reserve slice the fees that stores must pay banks each time a customer swipes a debit card, handing merchants a victory over banks in a lobbying battle over billions in revenue.

Senators supporting the financial institutions’ efforts to head off the proposal fell six voters short of the 60 needed to prevail. The vote was 54-45.

The tally also was a triumph Sen. Richard Durbin of Illinois, the No. 2 Democratic leader who had muscled a provision into last year’s financial overhaul law requiring the Fed to offer a plan for limiting the fees.

Those charges now average 44 cents per transaction and mean $16 billion annually for banks and credit card companies, according to Federal Reserve data.

The Fed has proposed holding those fees to a maximum of 12 cents per swipe. By law a final rule must take effect on July 21. While it might still be changed, few expect it to differ dramatically from the current proposal.

Thirty-five Democrats and 19 Republicans voted to delay the Fed’s plan. Voting to let the rules take effect were 32 Democrats, a Democratic-leaning independent and 12 Republicans. Sen. Joseph Lieberman, I-Conn., did not vote.