Celebrated on the $50 note, Edith Cowan (Nationalist Party of Australia) was the first woman to be elected to a parliament in Australia - in Western Australia in 1921. This was just a year after the law in that state was changed to allow women to stand for election. 
In August 1943, a further 22 years after Cowan was elected, the first women were elected to the national parliament  when Dame Dorothy Tangney (Australian Labor Party) became Senator for Western Australia and Dame Enid Lyons (Liberal Party of Australia) was elected to the House of Representatives.
Interestingly, South Australia is both first to allow women to stand for election and last to have a woman elected after a wait of 65 years.

1894-1923 Voting and standing for parliament for non-Indigenous white women. 

Excluding Aborigines and natives of Asia, Africa or the islands of the Pacific (except New Zealand) in federal law. Colonial and then state provisions to exclude non-whites were not uniform.  

Rights to vote and rights to stand for election for white women in Australia prepared by the NSW parliament.  

The status of Australia as a pioneer of women’s suffrage has also been undermined by the country’s racial policies. Political rights in the federal elections (but not in all state elections) were restricted to the white population only, while Aboriginal people were not granted the right to vote until 1962.

It has been easy to emphasise the universality of suffrage in Finland against the backdrop of the restrictions imposed in New Zealand and Australia.

Some have thus argued that Finland was the first to achieve “genuine” democracy. After all, not only did Finnish men and women of all social classes become politically equal in 1906, but Finland also adopted a unicameral parliamentary system, which was exceptional in Europe at the time. Moreover, both men and women were granted the right to stand for election, and they actively exercised this right from the very first parliamentary elections.
— http://www.helsinki.fi/sukupuolentutkimus/aanioikeus/en/articles/international.htm
 

Resources and other comments about women in Australian politics

Government site: Australian women in politics

Clare Wright provides more context surrounding the struggle for suffrage in Australia, focusing on the story of Vida Goldstein in Birth of a nation: how Australia empowering women taught the world a lesson. Wright argues that we have perhaps fallen behind in areas of social leadership and it is time to address issues of gender equality with renewed vigour:

Perhaps, with the global challenges of the 21st century, Australia can reassert its erstwhile youthful exuberance and once again be proud to call itself a trailblazing leader – a nation where justice serves as the foundation of its moral constitution.

Vida is recorded as the first person in the world to stand for election in a national parliament in 1903. When Vida died in August 1949 at the age of 80, two women - Dame Dorothy Margaret Tangey and Dame Annabee Rankin (Liberal Party of Australia) held Senate seats and two women - Dame Enid Lyons and Doris Blackburn (Independent Labor) held seats in the House of Representatives.  In the Encyclopedia of women & leadership in twentieth-century Australia, it is noted that, in the end, Vida was disappointed by the pace of change. 

Returning to Australia in 1922 she was disappointed by what she saw as the political apathy and spiritual void of young women, who knew little of the sacrifices that had been made for their freedom. She decided to bow out of public life.

the election of other women to the parliament 

Figure 1: Percentage of women in the Senate and House of Representatives, 1943 to 2011 Source: Parliamentary Library

Figure 1: Percentage of women in the Senate and House of Representatives, 1943 to 2011

Source: Parliamentary Library

In the period between Vida's death and 1959 three more women entered the Senate and no new female members were elected to the House. The following table shows the numbers of new women entering parliament by election or the filling of casual Senate vacancies in the following decades:

Number of new females elected to Parliament from 1949

Number of new females elected to Parliament from 1949