The result means she equals her best-ever grand slam performance.
OÃ¢â‚¬â„¢Brien overcame Mayr 6-3 6-3 as she broke twice in the first set to race into an early lead. The second set proved a more difficult affair but she displayed steely nerve to eventually win the tie.
Her next opponent will be eighth seed Jelena Jankovic. The Serbian will present a much sterner test for the 23-year-old hailing from Hull.
Compatriot Elena Baltacha will join OÃ¢â‚¬â„¢Brien in round two after she battled to a victory in the final set of her match. The Brit overcame injury niggles and troublesome weather conditions as she beat a player ranked three places above her 6-4 3-6 7-5.
Meanwhile in the menÃ¢â‚¬â„¢s singles, Louk Sorensen of the Republic of Ireland created his own segment of history when he beat TaiwanÃ¢â‚¬â„¢s Yen-Hsun Lu 6-4 3-6 6-2 6-1 to march triumphantly into the next round.
Lu is ranked 173 places above Sorensen but the Irishman continued in the rich vein of form which had helped in progress past the qualifying rounds. He had already become the first Irishman to reach the first round of a main even since 1985.
Speaking to the press after the match he said: “It was a tough match. I started off very well and then I felt a little injury in my leg, but I got through this and played very well, so I’m very happy now.
“The atmosphere was great, there were a lot of Irish supporters and it was amazing.”
Sorensen will face 33rd seed John Isner in the second round and the giant American will be clear favourite to ease into the third round.
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BIOGRAPHY: Nemanja Matic
BIOGRAPHY: Jesse Lingard