Incumbent Mayor Bobby Bryant was the runner-up Tuesday night, but he was ahead of third-place finisher Michael Todd by just 31 votes, or about 1 percent, according to unofficial election results.
"It's too close still," Todd said. "We're going to wait this out and let the election department sort this all out. This is the perfect example of why every vote counts."
The outcome could change if there are any contested ballots in the race. Tuesday's vote count will be formalized when votes are canvassed at a later date.
Meck said he felt confident about being able to beat Bryant in a runoff. Bryant, who could not be reached for comment late Tuesday, beat Meck in 2006.
"I feel like people heard my message, and they were willing to give me a chance," Meck said. "We're going to sit down Monday and plan out our attack from here."
Though Meck received 41 percent of the vote, he didn't earn enough to win outright. Candidates must get at least 50 percent of the vote to take office.
"I'm pleased that we won and we won with a pretty strong margin," he said.
In local council races, former Buckeye police Officer Brian McAchran barely missed winning the District 1 Town Council seat without a runoff, coming about two votes short of victory.
"There's always a chance something could change just a bit," said McAchran. "It really just means I carry on doing the same thing I've been doing."
He will run against Rick Fernandez, a small-business man, in the runoff May 20.
Incumbent Town Council members Robert Garza, of District 2, and Elaine May, of District 3, were voted in unopposed.
Buckeye residents overwhelmingly passed Proposition 500 to override state spending limits that would have cut $200 million from the budget next year.
But voters were less decisive about Proposition 501, extending the mayoral term from two to four years.
About two-thirds voted to allow it, bringing Buckeye in line with most West Valley municipalities. About a fifth of registered voters turned out, much more than the previous election.
The mayor's race was hotly contested. Meck, a 66-year-old Buckeye native, ran on his extensive knowledge of the town, while Bryant, who was hammered at campaign forums for fiscal mismanagement, asked for more time to solve town problems.
Political newcomer Todd claimed to be a fresh start for Buckeye.
