The Senate last night joined the House in giving overwhelming approval to President Reagan's austerity budget for next year, marking a dramatic reversal of spending habits for social welfare dating to the New Deal.

Five days after the Democratic-controlled House gave its assent, the Republican-run Senate swept away the last vestiges of disjointed Democratic opposition and voted 78 to 20 to approve a $700.8 billion version of Reagan's budget for fiscal 1982.

It did so after modifying its earlier proposal to cut $7.9 billion from Social Security and other federal retirement programs, restoring $1.7 billion so that military and civil service retirees will not have to wait longer than other beneficiaries for cost-of-living increases in their pensions. l

The budget now goes to a House-Senate conference, which is expected to resolve relatively minor differences in time for final approval next week of initial spending targets for the fiscal year beginning Oct. 1.

The House budget totals $688.8 billion, with a deficit of $31 billion, smaller, at least on paper, than the Senate budget of $700.8 billion, with its projected deficit of $50.5 billion.

But most of the gap between the two budget drafts stems from different assumptions about how the economy will perform, with the House taking a rosier view than the Senate, especially about interest rates.

[In a related development, a head count by House Democratic leaders showed that 60 of the 63 Democrats who voted for Reagan's budget in the House last week do not intend to vote for his tax plan. Details on Page D8.]

Both budgets make deep and broad cuts in education, health, welfare, housing, employment and other social services, while proposing the biggest peacetime increase in defense spending in history, all as Reagan wanted.

Both leave room for Reagan's proposed tax cuts, which would cost $54 billion in 1982. But neither commits Congress to Reagan's specific tax plan, which calls for an across-the-board tax cut of 30 percent over the next three years, with big cuts in business taxes, too.

Both also anticipate a balanced budget by 1984, as Reagan has promised, although there is nothing to guarantee that this will happen. The Senate version specifically concedes that the balance hinges on further spending cuts that Reagan has yet to spell out.

In the Senate vote, most Democrats joined all but two Republicans, Lowell P. Weicker Jr. (Conn.) and Gordon J. Humphrey (N.H.), in approving Reagan's budget, although some of them had supported modifications along the way. Eighteen Democrats, mostly from the industrialized Northeast and Midwest, voted against it, while 27 Democrats, plus Harry F. Byrd Jr. (Ind-Va.), voted for it.

Commenting on the vote, Senate Budget Committee Chairman Pete V. Domenici (R-N.M.) said it "clearly indicates the Senate is prepared to change its spending habits," and added that "over the long run, we have started a trend toward fiscal sanity."

But Democratic Whip Alan Cranston (Calif.), parting company with Minority Leader Robert C. Byrd (D-W.Va.) to vote against the budget, said it would fail to end deficits by 1984.

Sen. Bill Bradley (N.J.), head of the Senate Democrats' economic advisory group, also denounced it, saying it "reflects and reinforces an untested economic theory that gambles with the only economy we have" and "threatens to mortgage our future to a throw of the dice."

The budget only sets spending targets, although the Senate previously voted to require committees to make more than $36 billion in program cuts, as did the House in its budget resolution. Leaders of both houses have pledged to enforce the spending targets, but some strains are appearing.

Only yesterday, qualms by Weicker and Robert T. Stafford (R-Vt.) forced the Senate Labor and Human Resources Committee to put off action on part of Reagan's proposal to combine separate health, education and social-service programs into block grants that would be administered by the states.

Republicans control the committee, 9 to 7, making the votes of Stafford and Weicker critical. Stafford, a moderate, said he would like to try the block-grant approach but is concerned about losing worthwhile programs in the process.

Citing plans to combine educational programs for the handicapped and the economically disadvantaged into one block grant, he said, "It is the potentially destructive competition between those two interests that I don't want to see occur." Many House members, especially Democrats, have voiced similar concerns.

But on the Senate floor yesterday the pro-Reagan tide swept everything in its path as Democrats tried without success, and without much unanimity, to leave some imprint on either the tax or spending side of the budget.

In one of its closest votes, the Senate rejected, 56 to 42, a proposal by Howard M. Metzenbaum (D-Ohio) to do away with $79 million in tobacco subsidies.

But it refused, 76 to 22, to go along with a Bradley proposal to put back about $1 billion in spending for social programs, with the spending to be offset by ending two tax shelters.

By an even more lopsided vote, 81 to 17, it brushed aside a proposal by Donald W. Riegle Jr. (D-Mich.) to shave $2.2 billion off the defense spending increase and shift the money to social programs.

The Senate also rejected, 53 to 34, a proposal endorsed by the Democratic Caucus to substitute a tax cut adopted last year by the Senate Finance Committee for Reagan's tax plan.

The only approved amendment, offered by Sen. Ted Stevens (R-Alaska) and endorsed by the Republican leadership, corrected what was described as an error in drafting the cutback for Social Security and other retirement programs that was approved last week.

The amendment would have the effect of giving government retirees their 1982 cost-of-living increase in March if Congress adopts the cost-of-living cutback. Reagan has proposed other Social Security cuts, which may be adopted instead.