(Adds comment from House speaker)

CHICAGO, May 30 (Reuters) - Chicago will be able to spread out state-mandated higher payments to its police and fire pensions after the Illinois Legislature on Monday overrode the governor's veto of a bill that became entangled in a political impasse.

The Senate voted 39-19 and the House voted 72-43 to undo Republican Governor Bruce Rauner's veto on Friday that the city claimed would lead to a $300 million property tax hike.

The bill gives Chicago short-term budget relief but will add to the city's big pension funding gap.

The override bolsters Democrats, who control the legislature, as they battle with Rauner over state assistance for Chicago and its public school system, which is seeking state money for its teachers' pensions. The political impasse had left Illinois without a complete budget 11 months into fiscal 2016.

Rauner called the bill "terrible policy," while Chicago Mayor Rahm Emanuel accused him of using the city as a political pawn.

Rauner's spokeswoman Catherine Kelly released a statement reiterating the governor's contention that the measure would end up costing Chicago taxpayers $18.6 billion over time.

The measure alters a 2010 state law that boosted Chicago's payments to its public safety workers' pensions in order to reach a 90 percent funded level by 2040. Under that law, Chicago's contribution will jump to nearly $834 million this year from $290.4 million in 2015, according to city figures.

The new law reduces the payment to $619 million and allows for smaller increases through 2020 than under the 2010 law. It also gives the police and fire funds until 2055 to become 90 percent funded. The police system is 26 percent funded and the fire system 23 percent funded.

Chicago's fiscal 2016 budget assumed the bill's enactment by lowering the city's contribution to police and fire pensions by about $220 million. The city council also approved a $543 million phased-in property tax to exclusively cover higher contributions to the two retirement systems.

House Speaker Michael Madigan was clearly delighted as he addressed reporters after budget talks with Rauner.

"I think it was interesting the governor had nothing to say about the override. I was raised not to cause embarrassment for people so I didn't raise it," Madigan said.

The city must still deal with underfunding problems for its municipal and laborers' retirement systems after the Illinois Supreme Court in March tossed out a 2014 law that cut benefits and increased contributions to keep the pension funds from insolvency.

(Reporting by Karen Pierog; Additional reporting by Dave McKinney in Springfield, Illinois; Editing by Richard Chang)