The Republican-controlled Senate joined the Democratic House last night in giving final approval to an $859 billion budget resolution for the next fiscal year that defies President Reagan on all major counts.

The vote was 51 to 43 in the Senate and 239 to 186 in the House.

The fiscal 1984 budget resolution, a bipartisan House-Senate compromise that Reagan has denounced and vowed to ignore in vetoing tax and spending bills that he deems excessive, calls for higher taxes and more domestic spending than he wants, along with cutting in half the rate of his proposed military buildup.

Although House approval came easily, largely along party lines, the Senate vote was complicated at the last minute by an unsuccessful effort by Finance Committee Chairman Robert J. Dole (R-Kan.) to reduce the budget's requirement for $73 billion in tax increases over three years.

Dole was defeated, 51 to 41, despite his bitter complaint that the Senate was trying to "give this dead cat to the Finance Committee" by calling on it to achieve most of the 1984 deficit reductions needed to keep the red ink total below Congress' target of no more than $179.3 billion.

On the Senate's final vote, 19 Republicans joined 32 Democrats in support of the budget, slightly more Democrats and fewer Republicans than voted for the Senate's original version, which passed by a one-vote margin.

As a congressional resolution, the budget takes effect without presidential signature. It not only sets tax and spending targets for fiscal 1984 but also requires that committees draft legislation within a month to enact the $73 billion in tax increases along with $12.3 billion in spending cuts for the next three years.

Dole said after the vote there is "no possible way" for the Finance Committee to come up with the required tax increases, raising doubts about whether the budget will be fully enforced.

Both Senate Budget Committee Chairman Pete V. Domenici (R-N.M.) and Sen. Lawton Chiles (D-Fla.), leaders of the long effort to move the budget through the Senate, gave Majority Leader Howard H. Baker Jr. (R-Tenn.) a major share of credit for approval.

Baker, who is always reluctant to break with Reagan and was described by aides and colleagues as anguished over the choice, voted with Dole to reduce the proposed tax increases and then, without comment, voted for approval of the budget.

His vote with Dole was "his way to restate support for the president's position, and his vote for the budget was "to keep the budget process alive," an aide said. "He will vote with the president to sustain his vetoes," regardless of whether the vetoed legislation conforms to the congressional budget resolution, the aide said.

As negotiated last week by a House-Senate conference, the budget resolution for fiscal 1984, which starts Oct. 1, provides:

* Domestic spending of $21.5 billion more than Reagan wants, including an $8.5 billion "reserve" for recession relief programs, such as health insurance for the unemployed and farm and home foreclosure relief, that only would be given full-fledged budget status if the programs are enacted.

* A defense spending increase of 5 percent after inflation, or half what Reagan proposed, although defense spending authority would still increase by $23 billion to $268.6 billion.

* Tax increases of $12 billion in 1984, $15 billion in 1985 and $46 billion in 1986, a total of $73 billion, well above the nominal increases proposed by Reagan for the next two years but in line with the $50 billion stand-by tax increase that he proposed for 1986. In all, Reagan called for $59 billion in tax increases over the three years.

* "Reconciliation" instructions to House and Senate committees to come up with legislation by July 22 to enact the $73 billion in tax increases, along with $12.3 billion in spending cuts from Medicare, federal retirement programs and a few other sources.

* A 4 percent pay raise for federal workers, delayed for three months from October to January, along with a six-month delay in cost-of-living increases for federal retirees, similar to a delay approved earlier for Social Security.

* A deficit of between $169.9 billion and $179.3 billion, depending on how much of the anti-recession reserve is approved, with deficits declining by roughly $40 billion by 1986. By comparison, the deficit in Reagan's budget would have been $171 billion next year under a recent recalculation by the Congressional Budget Office.

House approval of the budget compromise came over strong objections from Republicans who called it "political gluttony" and a "blueprint for economic disaster," especially in its provisions for increased taxes and more domestic spending than Reagan wants.

Democrats responded by accusing the Republicans of "standing by and throwing rocks," as House Budget Committee Chairman James R. Jones (D-Okla.) put it, referring to Republicans voting against the bipartisan conference compromise after refusing even to offer Reagan's own budget as an alternative.

The House approved the conference agreement by a larger margin than the 229-to-196 vote by which it passed its version of the budget resolution in March. This time 10 Republicans voted for it, compared with 4 in March. The Democrats lost 33, as against 36 in March.

Both House budget votes were a major victory for the Democratic leadership, whose gains in last year's congressional elections helped end the string of Reagan budget victories during 1981 and 1982 and put House Democrats in a strong bargaining position to negotiate a bipartisan compromise with the Republican-controlled Senate.

Among Washington-area House members, only Maryland Democratic Reps. Steny H. Hoyer and Michael D. Barnes voted for the budget resolution. Maryland's senators voted for it; Virginia's senators voted against it.