The Fairfax County Board of Supervisors on Tuesday approved a revised funding agreement for the second phase of Metrorail’s extension to Dulles International Airport despite some members’ misgivings that the project’s total costs are still uncertain and likely to rise.

The board’s vote amounted to final approval for a deal brokered this year by U.S. Transportation Secretary Ray LaHood in a bid to shave costs from Metro’s Silver Line from Wiehle Avenue in Reston to Route 772 in Loudoun County.

County officials said the new agreement should reduce the total estimated cost of the second phase by about one-quarter, from $3.8 billion to $2.8 billion, and trims as much as $757 million from the burden on drivers who use the Dulles Toll Road. The biggest savings came after the Metropolitan Washington Airports Authority, which is managing the entire $6 billion project, reversed its decision to put the Dulles Airport station underground.

The U.S. Department of Transportation, Loudoun County Board of Supervisors and airports authority have signed off on the deal.Virginia has also agreed to kick in $150 million, subject to approval by the General Assembly at its session next year.

Fairfax Board Chairman Sharon S. Bulova (D) congratulated County Executive Anthony H. Griffin and members of his staff for negotiating the deal and urged its approval.

“This strategy brings the cost of Phase 2 back down close to the original estimate and goes a long way to ensure this critical transportation project is completed,” Bulova said in a written statement. Jim Corcoran, president and chief executive of the Fairfax Chamber of Commerce, also hailed the agreement.

But some supervisors expressed concerns about the ultimate cost to commuters and taxpayers as Fairfax County assumes responsibility for building a Metro station on Route 28 and two parking garages. The parking garages are estimated to cost $188 million.

Supervisor Michael R. Frey (R-Sully) said that instead of cheering the savings, supervisors should acknowledge that the new agreement is proof that the project has ballooned beyond its original $2.5 billion price tag and that more of the cost has been shifted to local governments. Frey said the total cost of Phase 2 now appeared to be more than $3 billion.

“I ’m not exactly sure why we’re rushing into signing some agreement when we’re not exactly sure we know what it costs,” Frey said.

Supervisor Patrick S. Herrity (R-Springfield) moved to delay a vote on the pact until the funding could be studied further. He said he was primarily concerned about the project’s effect on Dulles Toll Road users.

But Bulova and other supporters said delaying the agreement would send a bad signal to their partners and potentially cause delays that further drive up costs.

“I think delaying until we’ve satisfied all the what-ifs, what-ifs is not going to serve us well,” she said.

The Board voted 8 to 2 to back the agreement, with Herrity and Frey opposed.

Under the agreement, Fairfax will assume no more than 16.1 percent of the cost of the Route 28 station and two parking garages, even if no new revenue can be found, county officials said. Loudoun County will assume 4.8 percent of the cost, the airports authority will be responsible for 4.1 percent, and 75 percent will be carried by Dulles Toll Road revenue.

As part of the deal, Fairfax will tap its top credit rating to borrow up to $315 million under the federal Transportation Infrastructure Finance and Innovation Act and use part of those funds to reduce the costs to toll road users. The county also has agreed to find additional revenue and savings through contributions from developers and other third-party sources, as well as through possible public-private partnerships.

The Dulles rail line will add 23 miles of track. The first phase of the project, from Tysons Corner to Reston, is under construction. It is expected to open late next year. The second phase is expected to be completed in 2017.

Fairfax officials estimate the county’s share for the entire Dulles rail project to be about $900 million, of which $730 million would come from voluntary tax districts.

Robert Whitfield, a member of the Dulles Corridor Users Group, said the county is moving forward with a project that will saddle taxpayers and motorists with rising costs and will ultimately draw many fewer riders than projected. “It’s a financial disaster,” he said.