Hannah Klugman, the 15-year-old who won the prestigious Orange Bowl junior title in December, took just 56 minutes to beat Croatia’s Petra Marcinko 6-2 6-2.
Klugman is bidding to reach the main draw of a senior Grand Slam for the first time, having competed in the juniors at Wimbledon last year.
She was impressive against world number 265 Marcinko, who is 358 places above her in the world rankings.
Sonay Kartal was also among the British winners, the 22-year-old coming through 7-6 (8-6) 6-4 against Suzan Lamens of the Netherlands.
Kartal will face China’s Wei Sijia next, while Klugman takes on Czech Linda Fruhvirtova.
American former world number 21 Amanda Anisimova breezed past Greece’s Valentini Grammatikopoulou 6-2 6-3 as she continues her return to the sport.
Anisimova took an eight-month break in 2023 after struggling with burnout, external and her mental health.
Players have to win three matches to secure a place in the main draw at Wimbledon, which begins on Monday.
There were eight British women in qualifying action on Tuesday.
Britain’s Emily Appleton was narrowly beaten 3-6 6-3 6-4 by France’s Elsa Jacquemot, while Amelia Rajecki fell 6-4 4-6 6-4 to seventh seed Arina Rodionova of Australia.
Turkey’s Ipek Oz saw off Mingge Xu 6-4 6-4 and Croatia’s Tena Lukas won 6-7 (6-8) 6-4 7-6 (10-8) against Katy Dunne.
Fifteen-year-old British player Mika Stojsavljevic lost 6-3 6-4 to Selena Janicijevic of France.
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