Demonstrators briefly burst through a police cordon before being driven back with tear gas, witnesses said.
Mr Moreno declared a state of emergency last week but has failed to end protests led by indigenous groups.
They are demanding an end to austerity and the return of fuel subsidies.
Petrol prices have soared by more than 100% since last Thursday when the subsidies were removed.
The unrest has forced President Lenin Moreno to move the government out of the capital.
Indigenous-led protests have toppled three presidents in the past few decades.
Amid the worst unrest in years, the embattled government has said it is open to mediation through the United Nations or the Roman Catholic Church.
Witnesses said protesters, some of whom were carrying wooden shields, broke through a security cordon around the National Assembly building on Tuesday. Once inside they waved flags and chanted slogans before being forced back by security forces using tear gas. No staff were in the building at the time.
Other government buildings in the capital were also attacked and damaged, local media reported.
Across Ecuador, two people have died in the recent disturbances and dozens more have been injured.
Later on Tuesday, Mr Moreno declared a curfew restricting movement near government buildings between 20:00 and 05:00 in a bid to quell the unrest.
The president has refused to resign, reportedly telling broadcaster Teleamazonas "under no circumstance" would he quit.
"I don't see why I should if I'm making the right decisions," he reportedly said.
President Moreno had earlier announced that he had temporarily moved government operations from the capital to the port city of Guayaquil.
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