Dart beat Bulgaria's world number 194 Elitsa Kostova 6-4 6-1 in 75 minutes to set up a meeting with American 15th seed Nicole Gibbs.
Qualifying was delayed by an hour on Tuesday and practice was temporarily suspended because of the air quality caused by ongoing bush fires, but several players reported suffering breathing difficulties.
Slovenia's Dalila Jakupovic was forced to retire at 6-5 5-6 against Switzerland's Stefanie Voegel, the world number 180 being helped off the court after collapsing during a coughing fit.
"I was really scared that I would collapse. That's why I went onto the floor because I couldn't walk any more," Jakupovic said in her post-match press conference.
"I don't have asthma and never had breathing problems. I actually like heat. The physio came again and I thought it would be better. But the points were a bit longer and I just couldn't breathe any more and I just fell on the floor.
"It's not healthy for us. I was surprised, I thought we would not be playing today but we don't have much choice."
Former Wimbledon finalist Eugenie Bouchard also left the court, complaining of a sore chest, during her match against China's You Xiaodi, although the Canadian returned to play the final set following the medical timeout and won 4-6 7-6 6-1.
Smoke haze caused by wildfires led the organisers to temporarily suspend practice sessions for the tournament on Tuesday, with qualifying beginning later in the morning following a delay.
"Further decisions will be made based on onsite data, and in close consultation with our medical team, the Bureau of Meteorology and scientists from EPA Victoria," a statement from organisers said.
"As always the health and safety of our players, our staff and our fans is our priority."
Victoria's Environment Protection Authority had warned that air quality in the state – of which Melbourne is the capital – would range from moderate to hazardous because of wildfires that have been raging for months both in Victoria and the neighbouring state of New South Wales.
Australian Open tournament director, Craig Tiley, said last week that he was hopeful the tournament would go ahead but said air quality would be closely monitored.
Dart's victory was a rare success for British players on day one, with Liam Broady, Jay Clarke and Naiktha Bains all suffering defeats.
Broady was thrashed 6-3 6-0 by Ilya Ivashka of Belarus, while Clarke went down in three sets to Slovenia's Blaz Kavcic, a 32-year-old ranked 341st in the world. Bains lost 6-2 6-2 to Russia's Valeria Savinykh.