Arizona women had thier votes counted in the 1916 presidential election prior to national women’s suffrage in 1920.
The Arizona Republican published on election day (November 7th, 1916):
Woman's part in the presidential election is one of the outstanding features. In one quarter of the forty-eight states women have the vote this year. The twelve states are Wyoming, Colorado, Utah, Idaho, Washington, California, Arizona, Kansas, Oregon, Nevada, Montana and Illinois. The women in the last six states have the vote for president this year for the first time. The interest political leaders take in this vote can be appreciated when it is seen that the states involved have ninety-one electoral votes a little more than one-sixth of the electoral college's total vote.
Here’s a link to the newspaper: http://adnp.azlibrary.gov/cdm/compoundobject/collection/sn84020558/id/93742/rec/4
Additionally, Heidi Osselaer's Winning Their Place: Arizona Women in Politics, 1883-1950 provides a lot of valuable context for those years leading up to the 19th Amendment. Women is Arizona have always been tough but they faced especially unique challenges trying to organize their vote before the majority of the nation. And so early on in statehood, too!
Also, then-Congressman Carl Hayden’s mother, Nan Downing Hayden, who died in 1907, was a suffragette. She inculcated in her son support for the right of women to vote. Consequently, in 1913 Hayden introduced a joint resolution proposing a woman suffrage amendment. The resolution failed in committee but anticipated the Nineteenth Amendment by several years. Then, in 1918, Hayden sponsored the 19th amendment to the Constitution extending the right of suffrage for women. He was author of the language stipulating that rights enjoyed by women at the time of adoption would not be nullified or abridged as a result of suffrage. (In the same year, he was the sponsor and floor manager of the bill which established Grand Canyon National Park.). In 1927 he moved on to the US Senate where he served until 1969….served from Taft to LBJ. But I did cover his relationship w/ Carrie Chapman Catt and the Women’s suffrage amendment in my book, Vision in the Desert: Carl Hayden and Hydropolitics in the American Southwest (Ft. Worth: TCU Press, 1999).