The measures, including cuts salaries, were announced last week. Many public workers have not been paid in months.
On Monday the federal government froze Rio's accounts ordering the state to pay millions of dollars in unpaid debt.
The demonstrators, mostly firefighters, police and prison officers, spent three hours inside the assembly chamber.
During the occupation some wrecked windows, doors and offices.
Assembly Speaker Jorge Picciani called the protest "an affront to the democratic rule of law unprecedented in the political history of Brazil".
Both the state of the city of Rio de Janeiro have been struggling with a long-standing financial crisis because of a drop in global oil and commodity prices.
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The state declared a financial emergency ahead of the Rio Olympics earlier this year saying it did not have the funds to provide security for the Games and to finish a metro line.
Governor Luiz Fernando Pezao said that unless the austerity measures were approved by the state assembly, he could not guarantee that workers would receive their full salaries next year.
With tax revenues dropping, Rio has already made sharp budget cuts.
Mr Pezao said he would travel to the capital Brasilia to negotiate with the federal government.
Thousands of public workers have stormed the Rio de Janeiro assembly in protest at austerity plans to deal with the Brazilian city's financial crisis.