Britain’s Billy Harris came back from a set down to keep alive his hopes of qualifying for the Australian Open, while Ryan Peniston also progressed.
The 28-year-old Harris, who has never made the main draw of the men’s singles at a Grand Slam, recovered to beat Bulgarian Dimitar Kuzmanov 0-6 6-2 6-2 and set up a meeting with David Goffin.
Peniston won a final-set tie-break to beat Elias Ymer 6-1 4-6 7-6 (10-7).
But Lily Miyazaki was beaten on the first day of action in Melbourne.
Miyazaki lost to Australian 18-year-old Melisa Ercan, who was able to overcome calf cramps to edge a second-set tie-break and seal a 6-4 7-6 (7-5) win.
With the tournament playing catch-up after rain prevented any play on Monday, the other five British players in qualifying – Harriet Dart, Heather Watson, Francesca Jones, Jan Choinski and Oliver Crawford – had their opening matches pushed back to Wednesday.
They will each be hoping to join Harris and Peniston in the second of three qualifying rounds as they bid to reach the main draw of the year’s first Grand Slam tournament, which starts on Sunday.
Harris, Britain’s eighth-highest ranked men’s player, earned his first match win on the ATP Tour at the Sofia Open in November.
That breakthrough victory, on his ATP debut, ensured the world number 200 would be ranked high enough to play in the qualifying rounds in Melbourne.
Harris, who began his career driving between European tournaments in a van that had a bed in the back, is scheduled to face Belgium’s former ATP top-10 player Goffin – the current world number 107 – in his second qualifying match on Wednesday.
Peniston, 28, will meet Tunisian Aziz Dougaz after finally taking his eighth match point against Sweden’s Ymer, having missed four chances in the 10th game of the decider before producing a strong finish in the tie-break.
British number three Dart faces Italy’s Nuria Brancaccio, Watson takes on Swiss player Lulu Sun, and Jones plays Margaux Rouvroy of France in the women’s draw.
In the men’s, Choinski opens against Lebanese-German player Benjamin Hassan, while Crawford, who recently switched from representing the United States, faces Belarusian Ilya Ivashka.