The Miami Dolphins brought their big guns to Tallahassee on Monday, hoping to persuade the fickle Florida Legislature to approve a taxpayer-supported stadium upgrade in the final days of this year’s session.

In meetings Monday with top state lawmakers, National Football League Commissioner Roger Goodell, team owner Stephen Ross and team CEO Mike Dee made their last-minute pitch, pressing them to support an upgrade of Sun Life Stadium.

By day’s end, the NFL firepower succeeded — at least with Legislature’s Upper Chamber. The Senate approved a sports team bill in the equivalent of a legislative rout, 35-4 vote. The only no votes: Sens. Rene Garcia (R-Miami), Anitere Flores (R-Miami), John Legg (R-Lutz) and Senate President Don Gaetz (R-Niceville).

Despite the big victory, the Dolphins still face major hurdles in Tallahassee, where the House must approve a sports package before the full Legislature officially adjourns on Friday. Gov. Rick Scott also must sign off on the legislation.

House Speaker Will Weatherford has expressed concerns about the Senate bill, which offers up to $13 million annually in new tax breaks for sports teams.

Goodell, following his meeting with the governor, said final approval by Scott and the Legislature, along with Miami-Dade voters, would “send a strong message” to NFL owners, who meet next month to decide the host cities for upcoming Super Bowls. Goodell warned, however, he could not guarantee Miami would land a Super Bowl even if the May 14 stadium referendum passes.

“Let the voters vote and decide,” Ross said in an interview outside the Weatherford’s office. “This is a tremendous economic impact to Miami-Dade County and we’re just asking to allow the voters to vote,” he said.

In Miami-Dade, voting was already underway on Monday on the first day of early voting. Actual voters, however, were few and far between as only 1,531 cast ballots at 20 polling stations from Florida City to Miami Gardens. Many more voters — nearly 13,000 — returned absentee ballots to the county’s elections department in Doral.

“We remain optimistic and confident,” Dolphins campaign spokesman Eric Jotkoff said.

In all, about 184,000 voters had requested absentee ballots. The deadline to request absentee ballots is May 8. Miami-Dade has 1.27 million registered voters.

Some voters received absentee ballots because they automatically requested them for the next two general elections after last November’s presidential election. Others learned of them through info sessions the Dolphins had with the community, or postcards the team sent during its lobbying frenzy the past few weeks.

The Dolphins could receive up to $289 million from increasing the hotel tax and from a $90 million state sales tax subsidy, both over 30 years. The football club has agreed to refund the county up to $120 million and the state $47 million when the contract term expires in 30 years.

Ross has guaranteed several Super Bowls, college football championship title games, and international soccer matches here if the referendum passes. If the events don’t take place as promised he has agreed to pay substantial financial penalties to the county.

As the polls opened Monday, the first sign of an opposition campaign against the publicly-funded renovation plan was taking shape.