The four members of the five-seat body endorsed the special election, which board president Ken Yeager said would cost the county about $1 million for precinct and mail balloting.
The board will hold a second vote on the issue at its meeting next Tuesday before preparations can be set in motion.
District 5 Supervisor Joe Simitian said he wants to hear from people in Shirakawa's District 2 about the election at the public meeting.
The board vote came at a special meeting four days after Shirakawa resigned from office and agreed to plead guilty to charges of perjury, misuse of public funds and filing inaccurate campaign and finance statements.
As part of a deal with prosecutors announced Friday, Shirakawa also agreed to pay a $50,000 fine and repay the county $6,000 in expenses he charged to his county credit card.
Shirakawa was re-elected to his District 2 seat last November.
At the meeting today, County Executive Jeffrey Smith said that under the county charter, the board could appoint someone who lives in District 2 to fill the empty seat until a special election is held.
None of the supervisors favored appointing someone to the seat, however, opting instead to leave the post vacant until the election.
Under the charter, the county would have until June 4 at the earliest and June 25 at the latest to hold an election, Smith said.
A runoff election would be necessary if no one received more than 50 percent of the vote, and the runoff would have to be held within 56 days after that at a cost of another $1 million, Smith said.