Konta prevailed 7-6 (7-4) 2-6 7-6 (10-7) against the well-backed Australian and will now Garbine Muguruza after the Spaniard beat Zheng Saisai. She is the third Brit in round two, joining Katie Boulter and Dan Evans after a topsy-turvy match.
In 32-degree heat, the British No 1 got off to a poor start as Tomljanovic threatened to race into a 3-0 lead, but having shown great battling qualities, she took advantage of an unforced error after a fine cross-court backhand in a first-set tie-breaker to edge ahead.
The Brit, who was beaten by her opponent in Brisbane at the start of the month, showed signs of mental fragility in the second set when she was broken for 2-4, one wild forehand volley particularly errant, as an inspired Tomljanovic forced a decider.
Konta returned to court after a lengthy comfort break ahead of the third set, and she continued to struggle against the Australian No 3's serve.
But in the face of a partisan home crowd, the 27-year-old did enough to set up another final-set tie-break similar to that which saw fellow Briton Boulter come through in her match on Monday.
A 2-0 lead evaporated when several dropshots failed to pay off to hand the momentum to Tomljanovic but Konta regained her composure to get back to 7-7 before a cross court forehand winner sealed a dramatic win.
"Overall, every aspect of the match was tough," Konta said after the match.
"I played a tough opponent who I lost to recently [in Brisbane] and the first round of Grand Slams are never easy. And the condition were tough, humid and hot.
"But I competed hard and I'm happy I could compete on every single point.
Konta also revealed she had to remember that the 'super tie-break' is first to ten points, not seven. She added: "I was very conscious to not get excited if I got to seven - I saw Katie Boulter celebrate and thought that would be me," she joked.
"I was thinking 'if you get to seven you haven't won and just keep going'. "It is different, especially in a singles context, but it is the same for everyone."