President Carter's proposed department of education squeaked through the House Government Operations Committee yesterday, 20 to 19.

Although the margin was slight, the victory for the president was substantial. The House committee was considered the biggest obstacle to Carter's plan to group major educational functions in a single new department with Cabinet status. The Senate passed the bill earlier this week, 72 to 21.

Just before the final vote, Rep. L. H. Fountain (D-N.C.), who voted for the bill last year but switched yesterday, predicted that the new department, if it passes the House, will become "another monster in the federal government" and "it's going to grow and grow and grow."

Opponents, led by the American Federation of Teachers (whose rival, the National Education Association, is one of the chief supporters of the bill), will seek to kill the measure when it reaches the House floor.

Although Fountain and Rep. William S. Moorhead (D-Pa.) switched positions from last year, Chairman Jack Brooks (D-Tex.) drove the measure through the committee after a day of rapid amendments by picking up the votes of three Democratic freshmen and one GOP newcomer.

Before passage, the committee voted 17 to 11 to add Indian education programs to the new department, and by voice vote also added vocational rehabilitation. As approved by the committee, the new department would have an annual budget of about $14.2 billion and personnel of about 22,000, including 6,500 teachers in Indian and native-American schools.

After the vote, a spokesman for President Carter said, "We have now passed the toughest hurdle in our efforts to establish a separate department of education."

The department of education is the president's major government reorganization proposal this year. Brooks said he is ready to go to the Rules Committee to clear the way for floor debate as soon as possible, but just when it will reach the floor will be up to the speaker.

With the committee split just down the middle, tension rose as the committee took up one amendment after another and approached its final vote. With lobbyists scurrying about the room, attention focused on how the committee freshmen would vote and on the intention of two senior Democrats, who had voted for the bill in committee last year but were thought to be wavering.

One, of them, Moorhead, revealed his plans at Midday, when he said he had come to fear "centralizing the control over ideas" and therefore would oppose the bill.

The other, Fountain, whose state is engaged in a major struggle with the Department of Health, Education and Welfare over desegregation of the University of North Carolina system, tipped his intentions just before the final vote.

The defection of these two Democratic elders left the president with a base of 15 Democrats and long-time backer Frank Horton (R-N. Y.) on the 39-member committee. But Democratic freshmen Mike Synar (Okla.), Robert T. Matsui (Calif.) and Eugene Atkinson (Pa.) and Republican Lyle Williams (Ohio) also voted yes and put the bill across by one vote.

Carter has said the bill would give education better efficiency in management of federal grant programs, a bigger voice in government councils and a clearer focus, without taking over local control.

However, John N. Erlenborn (R-Ill.), who led the opposition yesterday, voiced fears of conservatives when he said the bill could allow federal domination of educational policy which "historically . . . has been [made] at the local level."

Some higher education groups fear that it would allow federal control of colleges, and the AFL-CIO has said creating a new department could break-up the power of the civil rights lobbying coalition.

Other labor groups, such as the United Auto Workers and State, County and Municipal Employes, back the bill, as does the NEA, the National School Boards Association and the Urban League.

Many believe a major factor in the dispute is AFT fear that the new department would be dominated by the rival NEA, to whom Carter initially made the commitment to seek a separate department.

In yesterday's vote, 18 Democrats and two Republicans backed the bill, while seven Democrats and 12 Republicans opposed it. CAPTION: Picture, REP. L. H. FOUNTAIN ..."It's going to grow and grow ..."