The vote was 13-2, district spokeswoman Mary Currie said.
The district's Finance Committee approved the plan Thursday by an 8-3 vote.
The district said two-thirds of bridge users now pay the toll using FasTrak and only 33 percent pay cash.
Collecting tolls electronically will save $19.2 million over 10 years starting in 2012, according to the district.
Eliminating 28 full-time and four part-time toll collectors and two vault employees will save $16.3 million in wages and benefits. The base salary of a toll collector ranges between $48,672 and $54,080 a year, according to the district.
Currie said some of the toll collectors may find other jobs as other employees retire.
The district faces an $89 million deficit over the next five years. Converting to electronic toll collection was among 33 initiatives approved by the board in October when the projected five-year deficit was $132 million.
Bridge users who do not use FasTrak will still have their license plate read. They will be sent a bill for the toll or they can pay it at kiosks in Bay Area stores. They may also pay with a credit or debit card over the phone or on the Web.