The 19-year-old Andreeva had her opponent on the ropes when Potapova went down with an apparent leg injury late in the match, putting a disappointing end to an otherwise competitive affair.
"She's my good friend and seeing what a battle we had... seeing her fall like this, it's not easy," said Andreeva. "It's not the way obviously you want to finish the match... I just told her she's been playing amazing so far."
The Roland Garros winner will next play the winner of an all-American clash between Iva Jovic and Coco Gauff.
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Potapova put up less than half as many unforced errors as her opponent in a tightly fought opening set, where Andreeva could not hang on when it counted the most, helping her opponent to the critical break in the final game with two double faults and a misfiring backhand.
But the Russian cleaned up her form considerably in the second set, thriving in the shorter rallies before Potapova handed over the decisive break with a double fault in the seventh game.
Andreeva broke her opponent from the baseline to go up 2-1 in the final set, where Potapova collapsed onto the court clutching her left knee in pain after chasing a ball along the baseline in the ninth game. Andreeva ran to her aid and helped her limp to her bench.
She lay on the court for several minutes as a physio taped up her leg and valiantly finished the match, as Andreeva clinched it with a backhand winner before going to the other side of the court to embrace her opponent.
"She's a better fighter than me so I'm sure she's going to come back even stronger," said Andreeva.