The Fairfax County Board of Supervisors approved a tax cut yesterday that will lower real estate taxes on the typical Fairfax home for the first time in more than a decade.

The board voted unanimously to lower the current real estate tax by 7 cents to $1.39 per $100 of assessed value. The move will trim about $29 from the real estate tax bill of the average Fairfax residence, which county officials say is valued at $105,160.

"This is the first time in memory that the average homeowner will be able to pay less in real estate taxes than he did the year before," said Supervisor Thomas M. Davis II, a Republican. "It is a tribute to our economic development program."

At the same time, the board delivered a severe setback to plans for a new county government center west of Fairfax City. It voted 9 to 0 to remove $14 million from the budget for preliminary road and sewer work at the site near Fair Oaks Mall. Supporters of the center said the action will not kill the center.

The board was determined to enact what Springfield Supervisor Elaine McConnell, a Republican, called "a meaningful tax cut" -- greater than 5 cents per $100 of assessed value. A cut of that size would have held taxes on the typical Fairfax home at their current level of $1,491 because of a 3 percent increase in residential assessments. Before this year's reassessments, the typical home in Fairfax was valued at $102,090.

Fairfax officials said that the tax cut, which had been expected, brings the county's real estate rate below the projected tax rates for the City of Alexandria and Prince William County, as well as below the tax burdens faced by residents in the District of Columbia and Montgomery County. The Fairfax rate, however, remains higher than Arlington's rate.

Despite the tax decrease, many Fairfax residents may find their overall payments to the county holding steady next year because of an increase in sewer rates that will boost an average household's water bills by $32 next year.

County Executive J. Hamilton Lambert discounted the sewer rate increase. "If the tax rate had stayed the same," he said, "the people would have paid $65 more" in real estate taxes because of higher assessments. "We're coming down in the metropolitan area, that's the critical thing."

There were few surprises in the three hours it took the supervisors to mark up the budget for fiscal 1986, which begins July 1. The county has been running a surplus for more than a year, largely as a result of the booming economy, new building and higher automobile sales. Supervisors had promised last year that a large tax cut was coming if the county coffers continued to swell.

The nine-member county board will make the tax rate final and adopt a budget of $1.25 billion at its next session, April 29.

The governmental center, which would be located on county-owned land at Rte. 50 and I-66, has been hotly debated by county politicians for almost a decade. While some have called a new structure necessary to avoid skyrocketing rental costs for offices now scattered around the county and in Fairfax City, others have derided the proposed complex as a "monument to bureaucracy."

When Lambert included $14 million for the complex in his proposed budget in February, it was an early target for supervisors eying a tax decrease, even those who have supported the government center in the past.

"Sooner or later Fairfax County government is going to have to move from the City of Fairfax," said Board Vice Chairman Martha V. Pennino, a Democrat. "Either we do it sooner and we pay within today's dollars or we do it later and we pay a great deal more."

Also eliminated from Lambert's budget was $2.3 million from a contingency fund set aside nearly three years ago for construction of the Dulles Toll Road. That road is open and being run by the state. The board also voted to apply a $4.4 million surplus from this year's budget to finance the real estate tax cut next year.

Supervisors asked Lambert to find an additional $500,000 they estimated was necessary to slash the real estate tax rate by 7 cents.

The board rejected bids by three supervisors to spread the decrease between the real estate tax rate and other taxes, such as the personal property tax, which primarily taxes vehicles, and taxes on electricity and gas.

Supervisor James M. Scott, a Democrat, urged a utility tax cut, which he said would benefit all residents. He said the real estate tax cut will not help renters in Fairfax, who comprise about 30 percent of the county residents. Scott's motion failed on a 7-to-2 vote.

Questioned by reporters after the board meeting, County Board Chairman John F. Herrity, a Republican, dismissed suggestions that the tax cut was a token. "The average homeowner's bill is going to be less -- that's all you can say," he said. "Less is better than more when you're talking about taxes."