Despite losing the first set to Alexandr Dolgopolov, Groth recovered to claim the next two sets before his Ukrainian opponent retired injured with the scoreline 4-6, 6-1, 7-5.
Groth will now meet Spaniard Tommy Robredo in the second round.
The news wasn't as good for the four other Aussies competing on day one with Daria Gavrilova, Casey Dellacqua, Matt Ebden and John Millman all suffering defeats.
Gavrilova was humbled by Russian teenager Daria Kasatkina 6-2, 4-6. 7-5 - a player ranked almost 100 places below her.
It was a disappointing result for Gavrilova after her side of the draw opened up after Maria Sharapova's withdrawal saw her pitted against the Russian lucky loser.
Dellacqua couldn't repeat her fourth-round heroics of last year, beaten 7-5, 6-2 by Estonian qualifier Anett Kontaveit.
Bulgarian 17th seed Gregor Dimitrov knocked out Ebden in straight sets 6-4, 6-2, 6-4 while Millman lost 6-1, 3-6, 7-6 (3), 6-4 in a tough four-set match against Ukraine's Sergiy Stakhovsky.