Johnson achieved worldwide recognition when, in 1930, she became the first woman to fly solo from England to Australia. Flying G-AAAH Jason, she left Croydon Airport, Surrey, on 5 May and landed at Darwin, Northern Territory on 24 May 11,000 miles (18,000 km). Six days later she damaged her aircraft while landing downwind at Brisbane airport and flew to Sydney with Captain Frank Follett while her plane was repaired. Jason was later flown to Mascot, Sydney, by Captain Lester Brain.
In June 1930, Johnson's flight to Australia was the subject of a contemporary popular song, Amy, Wonderful Amy, composed by Horatio Nicholls and recorded by Harry Bidgood, Jack Hylton, Arthur Lally, Arthur Rosebery and Debroy Somers. She was also the guest of honour at the opening of the first Butlins holiday camp, in Skegness in 1936. From 1935 to 1937, Johnson was the President of the Women's Engineering Society.