The Senate, meeting in an extraordinary Sunday session, failed yesterday to choke off debate on legislation to balance the federal budget by 1991, although the proposal won enough support to pass if further procedural hurdles can be scaled.

The Sunday session -- the fourth in 12 years -- was called because the budget proposal is being considered as part of debt-ceiling extension legislation that the Treasury Department says is needed by today to prevent disruption of government financing operations.

But the government will operate as usual today and probably longer if history is any guide. Moreover, Senate Majority Leader Robert J. Dole (R-Kan.) said he hopes to meet this morning with other congressional leaders to discuss ways to "avoid a crisis."

After coming into session in midafternoon with all but five members attending, the Senate voted 57 to 38 yesterday to limit debate by invoking cloture on the budget proposal. But this was seven votes short of the two-thirds -- 64 -- that would have been needed for cloture.

It would be more than a majority, however, for passage of the legislation itself. The measure picked up two more votes on a second procedural test.

Democratic leaders continued to hold out against quick action on the measure, protesting that it was so complicated and fraught with potential difficulties for the economy and operations of government that more time was needed for consideration.

The proposal would set fixed deficit limits over the next six years to force a lowering of the deficit from $180 billion annually down to zero by fiscal 1991.

If limits are exceeded, the president would be required to cut spending proportionately.

In a statement from his retreat at Camp David, President Reagan reiterated his call for prompt action on the measure.

"Zero hour is approaching," he said, adding, "The American people have grown very weary of delays, excuses and inaction."

It was unclear when, or if, Dole and other Republican leaders would be able to shut off debate. Cloture normally requires 60 votes. But a two-thirds majority is needed for legislation involving Senate rules changes, as the budget bill does.

"We established that we've got a strong basis of support" although the Democrats "established that we don't have the votes right now to invoke cloture," said Sen. Phil Gramm (R-Tex.), a chief sponsor of the budget proposal.

Democrats want a stopgap extension of the debt ceiling of 10 days or more while the budget proposal is considered.

Dole is insisting on a delay of no more than two or three days, apparently in hopes of using the deadline pressure to force approval of the measure.

Dole vowed to continue pressing for a quick vote.

"It's going to be high noon here before long," he said. He scheduled another cloture vote for today.

As for the debt measure, today is critical because the government's cash balances will be "virtually exhausted," according to Treasury Secretary James A. Baker III, and the government can borrow no more money.

The government cannot continue to borrow because it has hit the existing debt ceiling of $1.824 trillion, which would be raised to $2.078 trillion in the new legislation.

This, Baker observed in letters to Senate leaders, means "the United States could be in the position of defaulting on its obligations for the first time in history." At least in theory, its checks could bounce.

However, even if a deadlock continues, the government can continue to operate without serious disruption for some time, judging by past experience.

Offices will open today as scheduled, government checks will continue to be honored and Treasury officials said late Friday that federal employe's paychecks will be issued as scheduled today.

But the officials said this will be the last major government expenditure that can be made under the current debt ceiling.

Increasingly in recent years, Congress has been putting the government through debt "crises" as lawmakers pounce on the relatively veto-proof extension measure as a vehicle for their pet causes.

In the past, including a marathon debt-ceiling showdown in 1983, the deadline passed without serious disruption to the government's financial operations, except for postponing securities auctions, which has been done again this year. Ironically, this adds to government costs through loss of revenue from interest, and enlarges the national debt.

In the past, the Treasury usually has managed to find funds to limp along until Congress acted. The government also could get along for several days, at least, by tapping the borrowing authority of the government's Federal Financing Bank -- a course that Baker indicated he opposed but did not rule out.

Dole has said the situation could continue at least through this week, backing Congress against a more persuasive deadline: its plans for a four-day Columbus Day recess.

The Democratic-controlled House has been pushing for a stopgap extension of 15 to 20 days to tide the government over while lawmakers resolve their disputes, but Dole has resisted in hopes of forcing a decision by the recess.