Sibley, a Republican, is trying to retake the seat he held for 11 years and finished with the most votes. But with all precincts reporting, no candidate in the four-way race earned the required 50 percent of the vote to secure the seat.
The central Texas seat includes 10 counties stretching from Waco to the outskirts of Fort Worth.
Sibley, a lobbyist and former dental surgeon, captured 45 percent of the vote while Birdwell, also a Republican, came in second with about 37 percent. Birdwell, who survived the Sept. 11 terrorist attack at the Pentagon, trailed Sibley by more than 2,500 votes.
The winner of the runoff, which hasn't been scheduled, will serve out the remainder of the term left open by the resignation of Republican Sen. Kip Averitt.
"During this election, I have focused on my life-long experience of fighting for our conservative values, and my desire to continue that fight," Sibley, who held the seat from 1991 to 2002, said late Saturday. "In the coming weeks, I will continue to travel the district to meet and listen to citizens' concerns."
Birdwell said he was "thrilled with the outcome" of the election.
"The voters sent a strong message that this seat does not belong to lobbyist David Sibley and the special interests, it belongs to the people," he said. "Despite being outspent more than 4 to 1, and starting this race only eight weeks ago, we closed the gap with each passing day."
Saturday's race also included Democrat Gayle Avant, a political science professor at Baylor University, and Republican Darren Yancy, who owns an insurance agency.
Averitt announced in January that he was dropping out of the race, but it was too late to get his name off the ballot and he won re-election over Yancy in March.
The candidate who wins the runoff would hold the seat through January, but would still have to run in the November general election to win a full-term.
Averitt is expected to withdraw from the general election, which would require local party leaders for the 10 counties in the district to select a Republican and a Democrat to be on the November ballot.
Two Dallas-area House seats had been part of the special election, but Democratic Rep. Eric Johnson and Republican Rep. Van Taylor, who both won their primaries, were declared elected without an election to fill the unfinished terms of their predecessors, said Secretary of State spokeswoman Ashley Burton.
The incumbents in those districts are former Rep. Terri Hodge of Dallas, who resigned as part of a plea deal in a Dallas corruption case, and Rep. Brian McCall of Plano, who stepped down to become chancellor of the Texas State University System.