The Democratic-controlled House handed President Reagan a stunning victory yesterday as it voted, 253 to 176, for a slashed-back $688.8 billion budget for next year that fully embraces his proposed tax and spending cuts.

Sixty-three Democrats joined a unanimous Republican joined a unanimous Republican minority in sanctioning what leaders of both parties called a historical reversal of course for American government, in effect a vote to dismantle or drastically cut back dozens of social welfare programs that the Democratic Party had built over the past half-century.

The vote, which Democratic leaders attributed largely to the popularity of Reagan and his budget-cutting crusade, also signaled that the House -- the only citadel of government under nominal Democratic control -- under nominal Democratic control -- really marches to the tune of Republicans and conservative Democrats.

Reagan, resting in the Lincoln bedroom of the White House when aides called him with the tally, hailed it as a "resounding victory" for the economy and bipartisanship in government. "When the people speak, Washington will now listen," he said.

In approving a revised version of Reagan's budget that actually cuts deeper into domestic spending than the president proposed, the House, with its 241-to-190 Democratic majority, implicitly rejected a Democratic alternative that would have saved about $7 billion worth of education, health, job training and other social programs.

But even the Democratic counterproposal, drafted by the House Budget Committee, would have given Reagan 75 percent of the cuts that he wanted, meaning he had won most of his fight before it started.

The budget not only sets austere spending targets for fiscal 1982; it then orders congressional committees to rewrite underlying legislation to cut programs under their control by $36.6 billion. That is a fraction less than the $36.9 billion in program cuts mandated last month by the Republican Senate as it voted on spending cutbacks even before taking up its own version of the budget. The Democratic budget would have cut spending year-by-year, rather than cutting programs.

As the House was voting, the Senate began action yesterday on its $699.1 billion budget proposal, which differs little from the House version.

The Senate voted, 72 to 24, to preserve the Legal Services Corp., which the administration wants to kill. The organized bar is fighting to save it, and the Budget Committee had refused to kill it.

Final congressional approval of a budget along the lines of the House-approved version appears almost certain, although intense guerilla warfare is then expected all summer over specific program cuts and the final outcome is expected to hinge on the president's popularity and economic conditions over the next few months.

Because both budgets also provide about $50 billion for a tax cut, Reagan is also assured th lattitude for the three-year, across-the-board cut in individual income taxes that he wants, although he is expected to have a much tougher fight over tax cuts than he had over the budget cuts.

"The horse that runs fast doesn't always run long," House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) told reporters before the budget vote in suggesting that Reagan faces Republican defections as well as Democratic opposition on his tax program.

As people in wheelchairs watched from a section of the House gallery reserved for the handicapped, a silent protest against proposed cuts in their programs, Republican and Democratic leaders spoke in epochal terms of the vote on the budget, as revised with White House blessing by Reps. Phil Gramm (D-Tex.) and Delbert L. Latta (R-Ohio).

"Let history show that we provided the margin of difference that changed the course of American government," declared Minority Leader Robert H. Michel (R-Ill.), calling the budget a "small step for Congress but a giant leap for the country."

O'Neill reflecting the importance of the vote by winding up the debate himself, saw the day in no less sweeping terms but from a starkly different perspective. "Do you want to meat-ax programs that made America great" or take a more cautious approach that would allow for a change of heart later, he asked. "You close the door on America in the Latta bill," he added dispiritedly.

Other Democrats, especially liberals, were even harsher in their criticism as they were forced to choose between two main alternatives that both made drastic cuts in social welfare programs while providing massive increases in defense spending.

Rep. Theodore S. Weiss (D-N.Y.) called it a "Drop dead, America" budget, and Rep. Phillip Burton (D-Calif.) likened it to the Gulf of Tonkin resolution under which Congress turned over enormous powers to the president during the Vietnam war.

"An economic crime," said Rep. John Conyers Jr. (D-Mich.).

Responding to Gramm's repeated contention that the Gramm-Latta budget "fires real bullets" at government excesses, Rep. Thomas J. Downey (D-NY.) said bitterly, "When they hit some of these programs, they are going to kill people."

But Republicans and their conservative Democratic allies described it as a long-overdue mandate from the people.

"The American people want tax relief and they want tax relief now," said Latta. "They want relief from inflation, and they want it now. The American people want action." They care nothing about economic assumptions and other sources of complaint from Democrats, he added. "They want action from this Congress."

Gramm conceded that the Democratic budget would have provided a "slowdown in the growth pattern" of government but added: "The American people don't want a slowdown in that growth pattern, they want it reversed."

Said Rep. Jack Kemp (R-N.Y): "You can't help the poor if the nation is poor."

Budget Committee Chairman James R. Jones (D-Okla.) and O'Neill questioned whether the budget cuts, coupled with the Reagan tax cuts, would achieve their stated purpose of bringing down inflation and interest rates and stimulating economic growth.

Brandishing newspaper clippings with headlines telling of rising interest rates and approaches on Wall Street, Jones said, "We've barely gotten into the masquerade ball and the masks are already off."

At his press briefing before the vote, O'Neill almost made it sound as though defeat would be a mixed blessing for the Democrats. "If the Republicans are unlucky enough to win today," he said, "I guess the monkey is off the Democrats' back. . . . They're the Reagan cuts.The deficit is going to be the Reagan deficit. . . . It's Reagan's inflation."

But Reagan, in his comments on the vote, called it an "historic moment of commitment: to a government that can both serve the people and live within its means."

Added Reagan: "For years, the American people have been asking that the federal government put its house in order. Today the people have been heard."

Having won his budget entirely on the basis of Democratic defections, he pointedly thanked both Democrats and Republicans and interpreted the vote as a victory for a "spirit of bipartisanship that we can build upon in the months to and years ahead."

Among Washington area members, only Rep. Michael D. Barnes (D-Md.) voted against the Gramm-Latta budget.

The House-approved budget envisions a $28.8 billion increase over current spending but $42 billion less than would have been required without change in existing policies and laws. It anticipates a deficit of $31 billion. It allows for a tax cut of $51.3 billion.

The Democratic budget totaled $713.6 billion, although it used less optimistic economic assumptions and the Democrats claimed that the spending totals woulds be roughly the same if the assumptions were used for both proposals. It included a $24.7 billion deficit, allowing for a $30 billion tax cut. CAPTION: Picture, House Minority Leader Michel gives "thumbs up" on $688. billion budget plan. By James K. W. Atherton -- the Washington Post