Boulter, who had gone into qualifying in Melbourne off the back of winning the ITF W60 Canberra final, was pushed to a deciding-set super tiebreak by Caroline Dolehide, which she edged for a 6-2, 3-6, 7-6 win.
Jodie Burrage, whom Boulter had beaten in that Canberra final at the weekend, was also victorious with a relatively straightforward 6-2, 6-4 win against Australia’s Elysia Bolton.
Burrage, though, struggled with her first serve, with just 47 per cent of them landing in across the two sets.
There were a trio of wins in all from the British women on Tuesday as Yuriko Miyazaki held off the challenge of Hailey Baptiste 6-4, 7-5, Miyazaki the more clinical in converting four of six break points in contrast to the American’s two from seven.
And Katie Swan, who had impressed during the United Cup for Great Britain, last week was the only British woman to bow out in qualifying during the day, losing 6-4, 6-3 to Frenchwoman Jessika Ponchet.
The hopes of the British men in qualifying, however, ended on Tuesday. Ryan Peniston, who had shone in the grasscourt warm-up events to Wimbledon last year, was defeated in straight sets by Canada’s Alexis Galarneau.
And then Liam Broady agonisingly missed out to Emilio Gomez 6-3, 6-7, 7-6, Broady losing the super tiebreak to the No4 seed 10-7.