Democrat Andrew P. Miller held a slim lead over Republican John W. Warner with more than half the vote counted last night in Virginia's U.S. Senate race.

In Northern Virginia, two liberal members of the U.S. House of Representatives, Joseph L. Fisher of the 10th District and Herbert E. Harris II of the 8th won their races against conservative Republican challengers.

Fisher defeated Arlington lawyer and lobbyist Frank R. Wolf and Harris beat John F. Herrity, chairman of the Fairfax County Board of Supervis.

In the Senate race, Miller, a 46 year-old former state attorney general, was trying to end a decade of defeat for Democrats in contests for governor and senator in Virginia. Warner, 51, is a former secretary of the Navy making his first race for elective office.

With about 58 percent of the state'svote tallied, Miller had a lead of 5,400 votes out of 740,000 east

In Arlington County, Shephen H. Detwiler, a conserative endorsed by the Republican Party, appeared likely to defeat Joe N. Pelton, endorsed by the Arlingtonians for a Better County (ABC), in a race for a seat on the county Board. A Detwiler victory would give Republican-backed candidates a majority of the five-member board for the first time in a decade.

Virginians also were voting on a proposal to legalize betting at horseracing tracks and Northern Virginians were supporting the measure in early returns. Statewide results were not available early last night because of a computer failure in Richmond.

The voter turnout for the Senate election pparently was heavier than most party and campaign officials had predicted. Miller was expected to benefit from a larger turnout.

The campaign was waged against the backdrop of a decade of defeats for Democratic candidates for governor, U.S. senator and president in Virginia and was seen by Miller and others as a "do-or-die" effort to pull his party back into the moderate-conservative mainstream of state politics.

Miller tried to take advantage of the controversial Senate term of Republican William L. Scott, whose retirement opened up the Senate position he and Warner sought. The Democrat often referred to Scott's 1972 upset of former Democratic senator William B. Spong as a "tragedy." He said his election would restore influence and respect for the state in the Senate.

Miller won his party's nomination over seven opponents at a convention in Williamsburg last June 10 after a campaign in which he contended that he was the only Democrat who was well known enough throughout the state to reverse the recent Republican tide.

Miller had been twice elected attorney general in 1969 and 1973, but his political career was given a jolt in 1977 when populist Henry E. Howell,a former lieutenent governor, narrowly defeated him for the Democratic nomination for governor.

Two months after Howell's loss to Republican Gov. John N. Dalton further deepened Democratic gloom in Virginia, Miller announced for the Senate seat. He was the favorite throughout the race for the nomination through the first three convention ballots. State Sen. Clive L. DuValII of Fairfax finished second.

Warner was awarded the GOP nomination by the state central committee on Aug. 12, 10 days after the party's first nominee, Richard D. Obenshain, died in a plane crash.

Obenshain was a former party chairman and the champion of conservative Virginia Republicans. Despite his long standing as a party leader, he was almost upset at the Republican convention by Warner, a relative newcomer to Virginia politics.

Warner waged a lavish preconvention campaign. He spent a record $574,000. including $491,000 of his own money, the most ever spent by a candidate himself in quest of public office in Virginia. In the end, Obenshain won by 37 Votes on the sixth ballot at a Richmond convention attended by 8,000 delegates and alternates. It was believed to be the largest political party convention ever held in the United States.

Warner continued his heavy spending the general election campaign. By the last week of the race, he had lent $321,000 to the election effort. His total spending as of Oct. 23 exceeded $800,000 and was more than double Miller's outlay. His loans helped finance a radio and television campaign that Miller called a media blitz.

The Democrat made an issue of campaign financing, saying thatWarner was "trying to win the election on the depth of his pocket."

As the campaign wore on, Miller also began to capitalize on a series of Warner misstatements that Miller said raised questions about the Republican's integrity.

Miller pointed to the revelation that Warner had contributed money to former president Richard M. Nexon's campaigns when he said he had not.

He cited Warner claims that he voted for Virginia's conservative, independent Sen. Harry F. Byrd Jr. in 1970 even though he contributed $1,000 to Byrd's Republican opponent.

And he often pointed out that Warner claimed he had not sought support from organized labor even though state AFL-CIO leaders produced notes of a private meeting in which Warner asked for their endorsement.

In addition to these apparent discrepencies, Warner's statements about his appointment and service in the Nixon-Ford cabinet also became a campaign burden.

Early in the race,Warner said in a television interview that he tried to slow down racial integration in the Navy when he was secretary. He later called the television station, WJLA in Washington, in an effort to change his answer on the tape, claiming that he had missed the word "integration," in the question. Still later, however, he defended his cautious approach to affirmative action programs in the Navy as prudent.

After it was disclosed that Warner was a moderate Nixon donor and that the family of his former father-in-law, Paul Mellon,was a major source of contributions for Nixon, Miller questioned whether Warner's cabinet appointment was a political payoff.

In addition to these clearly negative factors for Warner, there was always the question throughout the campaign of whether his marriage to movie star Elizabeth Taylor would help or hurt him.

It was countless joint appearances with Taylor on behalf of Republican candidates during the past two years that propelled Warner to a position to seek the nomination. But her seven marriages, British citizenship and Hollywood background raised eyebrows among some Virginia traditionalists already inclined to regard Warner as a washington celebrity and "jetsetter."

Nevertheless, prominent Virginia conservatives led by former governor Mills E. Godwin aided the Warner campaign. The Republican also drew heavily on the support of Gov. Dalton, who made numerous appearances for him, and on state Attorney General J. Marshall Coleman and incumbent GOP members of Congress.

Miller brought a number of Democratc senators into the state to campaign for him, including Senate Majority Leader Robert C. Byrd of West Virginia.

Miller, however, shunned help from President Carter, who carried every southern state except Virginia in 1976, and withheld a formal campaign role from Howell, his intraparty foe. While some of Howell's followers resented this treatment of their champion, Howell himself appeared at several party and labor union events for the Democratic candidate.

The major issue debates between the candidates turned on taxes and the federal role in education. Warner supported a major, across-the-board cut in federal taxes similar to the Republican's Kemp-Roth proposalin Congress. He often said the tax cut should be coupled with reductions in federal spending, but Miller charged that his proposal was "irresponsible."

Miller said any tax cut should be matched by "dollar-for-dollar" reductions in spending to avoid inflationary increases in the federal deficit.

In the end, both candidates expressed disappointment with the tax bill finally passed by congress. Warner called it a "hoax" and Miller agreed that it would not provide meaningful tax relief after the effects of inflation and an increase in social security taxes next year.

Both candidates favored "indexing," a technique that would keep taxes from rising as a result of pay increases that keep pace with inflation, and a constitutional amendment requiring a balanced federal budget except in time of national emergency.

On the education issue, Warner tried to characterize Miller's support for a federal Department of Education as a departure from Virginia's traditional opposition to expansion of the federal bureaucracy.

Miller said the only way to control waste in the Department of Health, Education and Welfare is to create three new departments to administer its health, education and welfare programs.

Miller also endorsed the goal of one-third funding of state and local school costs out of the federal budget. Warner said such a federal stake in education would erode state and local control over schools.

The Voting Rights Act, a law that subjects Virginia and 21 other states to federal regulation of elections because of past racial discrimination, also divided the candidates. Warner opposed renewal of the act, which expires in 1982. Miller tried to satisfy both black and white supporters by saying that he would try to apply the law to all 50 states, but vote against renewal if that effort failed, as most believe it would.