"I'm very happy, because it was really tough," Linette said.
"I knew that at some point she would come back and I just wanted to keep playing aggressive. And even though I missed some shots in the second set, I just tried to stay aggressive and somehow I came back."
Next up for Linette on Friday is seventh seed Donna Vekic, who beat Fanny Stollar 6-2 3-6 7-6(13) earlier in the day.
"I've played (Vekic) in practice a couple of times and she's an extremely competitive player," said Linette.
"I expect a really tough match tomorrow, and I just hope I can play as good as today."
Eighth seed Yulia Putintseva dispatched American Jennifer Brady 7-5 6-2 to set up a quarter-final against Svetlana Kuznetsova, who crushed Ana Bogdan 6-2 6-1.
China's Zheng Saisai broke Belgian Ysaline Bonaventure six times en route to a 6-4 6-4 win and will meet American Allie Kiick in the quarter-finals on Friday.
Belinda Bencic of Switzerland received a walk-over when Nao Hibino withdrew prior to their match with an injury. The sixth seed will face Andrea Petkovic in their quarters on Friday.
(Reporting by Rory Carroll; Editing by Peter Rutherford)
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