The case is being made alongside the Netherlands to the International Civil Aviation Organisation in a bid to bring Russia back to the negotiating table to pay reparations.
The flight from Amsterdam from Kuala Lumpur was shot down over eastern Ukraine in July 2014, killing all 283 passengers and 15 crew onboard, including 38 Australian citizens and residents and 193 Dutch travellers.
The attorney-general says Russia must be held formally accountable for the crime and its consequences.
"At the international level we are asking ICAO to suspend Russia's voting rights," Senator Cash told Sky News.
"ICAO is responsible for civil aviation standards and Russia will be voiceless on the international stage if the ICAO does this."
The families of the victims want Russia to be formally held responsible for the crime, she added.
Negotiations between Australia, the Netherlands and Russia over the MH17 incident broke down when Russia walked away from talks in October 2020 and did not return, despite repeated attempts from the other two nations.
The new proceedings may force Russia back to the negotiating table and even result in an apology or compensation, Senator Cash said.
"The feedback I have had is that the victims' families have welcomed this next step," she told the Seven Network on Tuesday.
"We have always said to them those 38 Australians we will, as your government, not stop until Russia is held accountable."
As part of the legal action, Australia and the Netherlands are seeking a declaration that Russia broke the international civil aviation convention, also known as the Chicago Convention.
Prime Minister Scott Morrison said Russia's refusal to take responsibility for the incident was unacceptable.
"(The) joint action by Australia and the Netherlands is a major step forward in both countries' fight for truth, justice and accountability for this horrific act of violence," he said on Monday.
"The Australian government will pursue every available avenue to ensure Russia is held to account so this horrific act never happens again."
The legal action will be on top of proceedings taken by the Netherlands against four suspects for their individual involvement in the downing of the flight.
Australia and the Netherlands will rely on "overwhelming" evidence the flight was brought down by a Russian missile over eastern Ukraine in an area that was under the control of Russian-backed separatists.
The countries say the evidence showed the missile belonged to Russia and could only have been fired by a Russian crew.
The legal action comes after multiple western nations instituted sweeping sanctions against Russia over its invasion of Ukraine.
"Russia's unprovoked and unjustified invasion of Ukraine and the escalation of its aggression underscores the need to continue our enduring effort to hold Russia to account for its blatant violation of international law and the UN Charter," Mr Morrison said.
Labor leader Anthony Albanese said the opposition would back to the legal proceedings.
"This joint action with the Netherlands is an important step forward in this fight," he said in a joint statement.
"The illegal and immoral invasion of Ukraine has shown us the contempt Russia holds for international law. It must be held to account for its shameful actions."
Foreign Minister Marise Payne brushed off concerns that legal action could exacerbate tensions with Russia in the wake of its invasion of Ukraine.
"Russia and Russia's actions are exactly what is exacerbating tensions," she said on Monday.
"Wholesale breaches of international law, complete violation of the UN (United Nations) charter, is rather more significant than just exacerbating tensions."
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