The Fairfax County Board of Supervisors on Tuesday approved an approximately $6.7 billion budget that raises taxes and fees to support increased spending on schools, employee salaries and some services such as public libraries and social agencies.

Reflecting cautious optimism about the recovering economy, the board voted 8 to 2 to increase general fund expenditures to $3.54 billion, a 4 percent overall increase compared with the fiscal 2012 spending plan. The board also agreed to increase the storm-water services fee and the property tax rate to raise nearly $10 million in extra revenue, at a cost of about $79 for the typical county household.

Two of the board’s three Republican supervisors — John C. Cook (Braddock) and Pat S. Herrity (Springfield) — cast opposing votes. Both said they welcomed the increase in employee pay but said the fiscal 2013 budget raises taxes too much for too many nonessential programs and marks a return to the county’s free-spending ways.

Supervisor Catherine M. Hudgins (D-Hunter Mill) also sounded as if she would vote down the budget, but from a different perspective. She argued that the county should do more for its neediest residents. Afterward, she said she would have supported a higher property tax rate and storm-water services fee.

After imposing a near-freeze on employee pay for three consecutive years, the board approved the 2.18 percent increase proposed by outgoing County Executive Anthony H. Griffin — and more. The board committed $16.5 million to resume granting merit raises and longevity steps for public-safety employees and a 2.5 percent increase for general county employees. Employees received a 2 percent pay increase in September but nothing more since fiscal 2009.

But the board also agreed to make money-saving adjustments in retirement benefits for future employees, such as increasing the minimum retirement age to 55, and instructed staff to draft possible revisions to county ordinances.

The board also sweetened Griffin’s proposal for school spending, agreeing to give Fairfax County Public Schools $1.85 billion, more than 52 percent of its general fund.

Griffin’s budget called for boosting overall spending by about 2 percent above the revised fiscal 2012 budget. He also suggested leaving the property tax rate at $1.07 per $100 assessed value.

With higher home values, that would have translated into a tax increase of about $34 for the typical household. Griffin also suggested raising the storm-water service rate by one cent, to 2.5 cents per $100 assessed value, or about $47 more for the typical homeowner.

On Tuesday, the supervisors shifted a half-cent of Griffin’s proposed one-cent increase in the storm-water service rate to the property tax rate. The property tax rate would therefore rise a half-cent to $1.075 per $100 of assessed value while the stormwater services rate would go to 2 cents per $100 from the current 1.5 cents. The shift is necessary because the storm-water services fee goes to a fund only for those expenses.

Herrity said Tuesday that the budget would impose too great a burden on taxpayers at a time when the county executive is predicting $100 million shortfalls every year at existing levels of spending. Among the luxuries the county can’t afford, he said, was $40 million for affordable housing for families that earn $120,000 a year or less.

Supervisor John W. Foust (D-Dranesville) lit into Herrity, saying the affordable-housing program was an essential program that helped the county’s neediest people.

Hudgins, in an interview, said she would have supported boosting the property tax rate to $1.08 and increasing the storm-water fee by half a cent to ensure that the county was helping its neediest residents. She expressed annoyance that the board could find $1.5 million for schools to install artificial turf playing fields but could not do something to reduce Head Start’s waiting list of 1,300 children.

The board also approved a package of budget guidelines for incoming County Executive Edward L. Long Jr. The guidelines include exploring ways to provide the school board with staff independent of the superintendent and the ability to conduct audits.

Cook voted against the budget guidelines because they omitted his proposal to direct about $27 million a year in proceeds from the county’s vehicle registration fee toward transportation.

The budget is scheduled for formal adoption by the board on May 1.