In a contest between two teenage stars, it was 17-year-old Gauff who sealed her second semi-final spot in as many weeks, having made the last four at the Internazionali d’Italia.
Third seed Gauff will face Katerina Siniakova next, after the Czech followed up her win over Serena Williams with a 7-5 6-1 defeat of Caroline Garcia.
Martic and Gauff move on to quarter-finals in Parma
“I was pretty satisfied with the way I played,” Gauff said, after fending off Anisimova.
“I do think I could have served a little bit better, but other than that, I was pretty happy with the way I played.”
Despite her valid concerns, Gauff won 72 per cent of points after landing her first serve and forced six breaks to win in style, reeling off six games on the trot from 3-0 behind in the second set.
The other semi-final will see second seed Wang Qiang take on Sloane Stephens.
Wang prevailed 7-6 (7-4) 3-6 7-5 against Petra Martic, while Stephens saw off Sara Errani 6-3 6-0.
It was another rain-affected day at the Serbia Open, where Maria Camila Osorio Serrano managed to get past Kamilla Rakhimova 4-6 6-3 6-4.
Anna Kalinskaya and Reka Luca Jani were locked at one set apiece in the only other match to get started before the weather halted play.
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