The American third seed arrived in Melbourne as the most in-form player on tour after winning the WTA Finals to end last season and then all five singles matches at the United Cup, including one against Iga Swiatek.
She went into the last-eight tie with 11th seed Badosa on a 13-match winning streak but it was the Spaniard who came out on top in a 7-5 6-4 victory.
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Badosa feared she may have to retire last year because of a persistent back problem but ended the season strongly, making the quarter-finals of the US Open, and has now hit a new high in Melbourne.
“Today I came in and I wanted to play my best game, and I think I did it,” she said. “Coco, at the beginning she was playing insane tennis, but I’m super proud of the level I gave today.
“A year ago I was here with my back and I didn’t know if I had to retire from this sport and now I’m here playing against the best in the world. I won today, I’m in the semi-finals. I will never think that a year after I would be here.”
Both players held serve until 5-5 in the first set, when Badosa struck before opening up a 5-2 lead in the second set.
She was unable to serve it out at the first time of asking but made no mistake the second time around to set up a clash against either top seed Aryna Sabalenka or Anastasia Pavlyuchenkova.
Gauff was left to rue 41 unforced errors but was philosophical afterwards, saying: “Paula was playing great. Maybe some moments in the first set could have gone my way, could have been a different outcome in the first set.
“I think it’s just a lot more work to do. I’m obviously disappointed, but I’m not completely crushed. I’m looking forward to a lot. Even though I lost today, I feel like I’m in an upward trajectory.”
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