The approval this week of Vienna's $6 million fiscal 1981 budget marked the end of Northern Virginia's annual battle of the budgets.

County and municipal budgets throughout the area, which take effect July 1, rose by an average of 15 percent over last year. Despite the increases, area officials maintain that Northern Virginia governments will be hard pressed to keep up with inflation.

Funds for schools, Metro and continually increasing energy costs accounted for most budget increases.

Salaries for county employes rose by 8.5 percent in Arlington and Fairfax and by 10 percent in Alexandria, the largest increase in recent years. County workers complained, however, that the salary hikes fell far behind the acutal cost of living.

In Fairfax County, a coalition of firefighters, teachers and other county employes warned officials at budget hearings earlier this year that an 8.5 percent increase was inadequate.

"The proposed increase would result in th largest loss to inflation which county employes have ever suffered in a single year," said Rick Nelson, president of the Fairfax County local of the American Federation of Teachers, when he testified at one public hearing.

One item that remained constant or decreased in most Northern Virginia jurisdictions was the property tax rate. For most homeowners, however, that will not mean lower tax bills. Because tax assessments have risen by more than 15 percent this year, most homeowners will actually have higher tax bills.

For example, the average assessed value of a home in Alexandria is $84,000. Even with the tax rate cut, from $1.44 for each $100 of assessed value to $1.39, that homowner will pay about $159 more in property taxes during the coming fiscal year.

School operating costs accounts for the lion's share of most Northern Virginia budgets, with Fairfax County spending the greatest proportion of its funds on education.

In Fairfax, 48.8 percent of the county's $442.7 million budget will go to schools next year, while Alexandria is allocating 38 percent of its $110 million budget for education. Arlington County comes in third among the "Big Three" Northern Virginia jurisdictions in school spending. Arlington has designated 29 percent of its $180 million budget for schools.

The increasing costs associated with Metro also are consuming larger slices of area budgets. Fairfax will spend $17 million on the transist system next year; Arlington, $9 million, and Alexandria, $5.8 million.

Perhaps the most difficult expenditure to pin down is for energy needs.

"Energy costs increase spending in every area," groaned an Arlington County budget official when questioned about the dollar amount the county will spend on energy next year.

"Energy costs are insidious, they're everywhere -- impossible to point to, and even harder to anticipate," said one Fairfax County official.