Congress gave final approval yesterday to President Carter's first attempt to reorganize the government as it sent him a bill creating a Department of Energy.

The House vote approving a compromise worked out at a House-Senate conference was 353 to 57. The Senate approved it by a vote of 76 to 14.

The legislation would pull together most federal agencies dealing with energy into a Cabinet-level department to administer the national energy policy Congress is attempting to write. Included would be the Federal Energy Administration, the Energy Research and Development Administration and the Federal Power Commission, and pieces of various other agencies. It will start off with a staff of 20,000 and a budget of $10 billion inherited from existing agencies.

James R. Schlesinger Jr., the President's energy adviser, is scheduled to be the department's first secretary. The Senate Energy Committee has scheduled hearings today on his nomination, though it has not yet been submitted.

The major change made by Congress in Carter's plan gives the power to set the price of natural gas to an independent five-member regulatory commission within the department rather than to the secretary.

Managers of the bill argued that the power to make these multibillion-dollar decisions was too great to be lodged in a single person and should be given to a commission insulated from political pressures.

As finally passed, the bill gives the commission power to set the price of new natural gas but authorizes the secretary to make proposals and to set reasonable time limits for the commission to make a decision.

The commission would also set wholesale interstate electricity rates and would share with the secretary limited pricing powers over domestic crude oil. (Oil prices were basically set by a 1975 law and under the pending bill would be taxed up to the world price.)

The power to set gas prices would be crucial to carrying out a coherent national energy policy if Congress approves Carter's proposal to continue price controls over new natural gas and extend them to intrastate as well as interstate gas.

The new Energy Department would acquire energy functions from four Cabinet departments and one independent regulatory agency.

From the interior department it would take regional marketing functions over electric power, now handled by the Bureau of Reclamation and Interior's four regional power administration; coal development and energy data programs, now in the Bureau of Mines; and control over the rate of energy production on the public lands, now in the Geological Survey and the Bureau of Land Management. The BLM would retain responsibility for actual leasing of public lands.

The new department would also be given:

Authority to set building conservation standards, from the Department of Housing and Urban Development.

Voluntary industrial conservation programs, from the Commerce Department.

Jurisdiction and administration of the naval petroleum and oil shale reserves in California, Wyoming, Colorado and Utah, from the Department of Defense.

Authority to set oil pipeline and coal slurry rates, from the Interstate Commerce Commission.

The House had written in a "sunset" provision requiring that the department go out of existence by the end of 1982 unless Congress extended its lift. The administration opposed this, arguing that it would prevent long-range planning. The conferees settled on a provision that merely directs the President to make a comprehensive review of the department's performance and report on it to Congress by Jan. 15, 1982.

The Department of Energy becomes the 12th Cabinet-level department, the first to be created since the Department of Transportation in 1966.