The Senate yesterday overwhelmingly approved $36.9 billion in budget cuts for next year as most Democrats joined the triumphant new Republican majority in handing President Reagan a resounding early victory for his economic progam.

Brushing aside complaints from some Democrats that the Senate was acceding to "scorched-earth" budget policies and a "cruel abandonment" of America's needy, the Senate voted, 88 to 10, to give Reagan all he wanted -- and then some -- in the first phase of his massive, unprecedented budget-cutting drive.

Senate passage of the spending retrenchment measure shifts the budget battle to the Democratic-controlled House, where Democrats believe they have a better chance of retaining money for educational, welfare, employment and other programs the Senate is cutting in accord with Reagan's wishes.

But the House battle was joined early by Rep. Delbert L. Latta (R-Ohio), ranking minority member of the House Budget Committee, who told reporters yesterday that he expects enough Democratic defections for a Reagan victory there, too.

Senate Budget Committee Chairman Pete V. Domenici (R-N.M.) predicted that the lopsided Senate votes for Reagan's proposals would have a persuasive influence in the House.

With the new Republican majority in total control of the Senate, Democrats lost repeated attempts to restore money for social programs that Reagan wanted to cut during six days of debate on the measure.

In the end, all but nine of them voted for the cuts. Among Republicans, only Sen. Lowell P. Weicker Jr. (R-Conn.) broke ranks.

The outnumbered, often-divided Democrats had to be content with speech-making and roll calls aimed at forcing Republicans to go on record against spending for politically popular programs ranging from veterans' benefits to child-immunization programs.

The Republicans hardly flinched, however, even voting without a single dissent against a Democratic-sponsored proposal to save $1.7 billion from "wage, fraud and abuse," a cause that many of them (and Reagan) championed when they, rather than the Democrats, were on the outside looking in.

And when some moderate Republicans defected, as they did in one significant case, the GOP majority could count on enough Democratic defections to win.

"Republican rigidity won the day," complained Minority Whip Alan Cranston (D-Calif.) in one of several harshly critical summations from Democrats yesterday.

"But the Republican victory may be a costly victory for the nation," he added, calling it a "cruel abandonment of America's commitment -- indeed America's obligation -- to help those most in need."

Said Cranston: "The Republican rule was don't change anything. Republican leaders, whipping their troops into lock-step conformity, refused any accommodation, any compromise that might have brought about truly bipartisan support for much of the president's budget."

Sen. Thomas F. Eagleton (D-Mo.) was even more acerbic. Likening the administration's budget proposal to a "scorched-earth" policy, he said it "asks the greatest sacrifices from those have the least to give [and] disavows good and bad federal programs alike in its quest for transient dollar savings."

He accused the Republicans of a "mindless show of one-upsmanship" in cutting even deeper than Reagan had proposed.

The Senate resolution exceeds Reagan's proposed cuts by $2.8 billion, although $500 million of it resulted from a largely bookkeeping change. It anticipates $36.9 billion in savings for fiscal 1982 and a total of $87 billion from 1981 to 1983.

It amounts to the first step in a long, complicated process: instructions to each legislative committee to cut programs under its jurisdiction by a specific amount, namely $36.9 billion. When the committees decide what programs to cut, the savings will be assembled in one big "reconciliation" package that then will be voted on by both houses.

Under a slightly different procedure, the House Budget Committee will start drafting its budget-cutting instructions next week. Final action on both the budget cuts and the spending targets for 1982, including tax-cut legislation, is planned by midsummer.

The speed with which Congress has responded thus far to Reagan's economic program has been extraordinary: little more than three weeks for the normally deliberative Senate. But the Senate, with its intensely loyal new Republican majority, has always been the easier hurdle.

Even Republican Latta conceded yesterday that opposition to the program was mounting in the House, at least before the attempt on the president's life Monday, and the House Democrats so far show more unity than their Senate counterparts, despite what is widely believed to be a pro-Reagan conservative majority in the House.

The Senate cuts affect nearly every category of spending except defense and so-called "safety-net" programs such as basic Social Security benefits, cutting most heavily into education, welfare, employment, nutrition, health and related programs.

Although Sen. Jesse A. Helms (R-N.C.) succeeded in shifting $200 million from foreign aid to nutrition, with the promise of an additional $100 million to come, Democrats failed in all attempts to restore other funds that were cut.

Among Washington area senators, both Virginians voted for the cuts. Sen. Paul S. Sarbanes (D-Md.) opposed the cuts, and Sen. Charles McC. Mathias Jr. (R-Md.) did not vote.

Meanwhile, the Congressional Black Caucus presented its "alternative budget" to the Senate Budget Committee, proposing to restore $25.8 billion in funds to aid the poor, close $27.1 billion worth of tax "loopholes" and wind up with a deficit slightly lower than Reagan's proposed deficit of $45 billion for next year.

The caucus would substitute a 10 percent tax credit on Social Security taxes, a higher standard deduction and some business stimulus tax cuts for Reagan's proposal of a 30 percent cut over three years.