Virginia voters, rejecting arguments that gambling is inconsistent with the character of the Old Dominion, approved a state-run lottery yesterday that could be competing with legal games in neighboring Maryland and the District as early as June.

Passage of the referendum, by 57 percent to 43 percent, automatically activates legislation on Jan. 1, approved by the General Assembly earlier this year, that establishes a state department of the lottery.

Gov. Gerald L. Baliles, who opposed the issue, said last night that "the people decided" and pledged to see that the lottery is administered "fairly, efficiently and honestly."

In Arlington County, incumbent County Board Chairman Albert C. Eisenberg and William T. Newman Jr., both Democrats, won the two at-large seats on the five-member board. Newman became the first black on the board since Reconstruction.

Prince William County voters chose a majority of newcomers on the seven-member board of supervisors to wrestle with the problems of growth that its neighbor to the north, Fairfax County, began encountering two decades ago. Three of four incumbents who sought reelection were returned, but Republican incumbent Guy A. (Tony) Guiffre lost to Democrat Robert L. Cole. Three other newcomers replaced incumbents who did not seek reelection.

In the only countywide race in Prince William, Commonwealth's Attorney Paul B. Ebert, a Democrat, defeated Republican Peter W. Steketee. Voters there also approved $44.89 million in school bonds.

In Loudoun County, two slow-growth proponents ousted a pair of Republican incumbents in contests where development and traffic were the principal issues.

Andrew R. Bird III was beaten by his ex-wife, Alice G. Bird, an independent, and Frank I. Lambert was replaced by Democrat Betsey J.S. Brown.

The lottery referendum was the only statewide issue on an Election Day in which Democrats retained their huge majorities in both houses of the General Assembly, largely because the GOP did not contest many incumbents. Where they did, Republicans mounted major challenges.

The Republicans' biggest victory came in suburban Richmond, where political novice Eddy Dalton, widow of former governor John Dalton, defeated the senior Democrat in the Senate, William F. Parkerson Jr., in the most expensive legislative race in Virginia history.

"We'll take it," state GOP spokesman Steve Haner said of Dalton's narrow margin, about 1,000 votes. "They spent a fortune, and we spent three-quarters of a fortune," he said of the combined spending of more than $1 million.

In Hampton Roads, incumbent state Sen. Democrat William T. Parker of Chesapeake was ousted by Republican lawyer Mark Early; and in the Richmond area, Democrat Del. N. Leslie Saunders of Chesterfield lost to businessman Stephen H. Martin.

Democrats, however, also retired some incumbent Republicans.

In Virginia Beach, Sen. A. Joseph Canada Jr., who lost a race for Congress last year, was defeated by Moody E. (Sonny) Stallings, a Vietnam veteran who got strong backing from former governor Charles S. Robb. In Danville, the Republican stranglehold in that Southside area was loosened when lawyer Whittington W. (Whitt) Clement defeated Del. Kenneth Calvert.

The lottery carried seven of the state's 10 congressional districts, trailing only in the largely rural western and southern areas.

Bob Weed, a D.C. political consultant hired by Virginians Against State-Sponsored Gambling, said opponents "obviously succeeded in taking people who were suspicious of a lottery and making them angry, on fire. We closed the gap {early polls showed the lottery favored by better than 2-to-1}. But those who slightly favored it didn't budge."

Ken Story, spokesman for Virginians for the Lottery, said the large turnout, and the result, proved that "people want to play a lottery."

In states where voters have been given a chance to vote on a lottery, the issue has failed only in North Dakota.

Among the big names who opposed the lottery, in addition to Baliles, were Attorney General Mary Sue Terry and former governors Robb, Linwood Holton and Mills E. Godwin. Leading newspapers throughout the state and hundreds of ministers also came out against the referendum.

The biggest financial contributor to the antilottery campaign, which amounted to about $375,000, was the United Methodist Church, which spent about $165,000 opposing passage.

More than 80 percent of the $210,000 spent boosting the lottery came from out-of-state companies that either would be involved in the daily operation of a lottery or would benefit by selling tickets. Heading the list of proponents were Scientific Games, a subsidiary of Bally Inc., and the Southland Corp., whose 7-Eleven stores would likely sell lottery tickets.

The well-financed antilottery campaign stressed that no state has ever reduced taxes because of lottery revenue. In states with the largest lotteries, including Maryland, taxes are one-third higher than in Virginia.

Opponents also pointed out that to meet revenue projections, Virginia would need half of the state's families to spend $600 a year on tickets.

Supporters contended that a lottery is a voluntary tax that would make an additional $200 million or more a year available for transportation, education and other projects.

One group that did not take a stand was the NAACP, which at its state convention Sunday rejected a motion to endorse the lottery and then passed a resolution suggesting that voters follow their conscience on the issue. Lt. Gov. L. Douglas Wilder also remained neutral.