Tomljanovic fell 3-0 behind before battling back to take the first set to a tie-break, where she held her nerve before rounding out the win in more convincing fashion in the second.
"Not the greatest start," said Tomljanovic. "I don't even think it was nerves. I wasn't too focused at the start.
"With her, you have to be really sharp, disciplined, but still aggressive. It took me a little bit to find the balance.
"But I think once I got going, it was a battle. I mean, she's always tough. She never gives up. She runs down every ball, really makes you win it. This is a quality win for me."
Another Australian in the draw is top seed Ashleigh Barty, who got a first-round bye and will now face Anastasia Pavlyuchenkova after the Russian's 6-2 6-4 win over Anett Kontaveit.
Fourth seed Belinda Bencic beat Daria Kasatkina 6-4 6-4, while Sofia Kenin defeated Viktorija Golubic 6-0 6-4.
Other first-round results saw former world number one Angelique Kerber down Wang Qiang 6-1 6-3, Dayana Yastremska beat Timea Babos 7-5 6-3, and American Danielle Rose Collins dump out Aliaksandra Sasnovich 6-3 6-2.
At the Hobart International, seventh seed Rebecca Peterson retired hurt against Fiona Ferro, but fellow seeds Magda Linette and Veronika Kudermetova progressed without cause for concern.