Dart, who is 22 years of age and the one who entered the 2nd round after coming through qualifying of the Brisbane International, defeat 6-3 4-6 6-2 Chloe Paquet of France.
After Gabriella Taylor and Naomi Broady missed their matches, Dart was the only British winner in Melbourne.
Dart, who is World no. 132 will next face China’s Xu Shilin as she wants to qualify for the first time in Melbourne.
Dart is three match away to reach the Australian Open’s main draw which is going to start on 14th of January.
Kristie Ahn from America defeated Taylor in just one hour and five minutes therefore Taylor missed the first 4 games of the match.
Belgium’s Ysaline Bonaventure defeated 6-4 3-6 6-3 British no. 7 Broady.
On Wednesday, James Ward, Katie Swan, Dan Evans and Jay Clarke start their qualifying campaigns.
Elsewhere, Sloane Stephens reached within 2 points of loss in her 1st round match against Ekaterina Alexandrova the qualifier at the Sydney International.
World no. 5 Stephens the American, won 0-6 7-6 (7-3) 7-6 (7-3) in the end.
Belarusian Aryna Sabalenka was defeated 6-1 7-5 by Czech Petra Kvitova, while Germany’s Angelique Kerber the defending champion defeated 7-6 (7-3) 6-2 Camila Giorgi of Italy.
However, Garbine Muguruza, former World no. 1, was retired ill from the event – the last WTA tournament before the Australian Open.