Former US Open champion Gauff, who has made some grip changes with new coach Matt Daly, struggled with double faults over both sets but was dominant with her returns in the victory over an error-prone Linette.
"I played well ... overall, maybe two bad games," Gauff said in the post-match interview after extending her winning streak to nine matches.
Linette came close to breaking Gauff twice in the first set but Gauff saved four break points to hold both games, also winning 14 of 21 return points as the Polish former Australian Open semi-finalist fell to a bagel.
Gauff, who made eight double faults in the match, faced a tougher battle in the second set, but one of Linette's 22 unforced errors gave the American an edge in the seventh game as she got the only break of the set, which sealed the win.
The winner of Friday's third quarter-final, between US Open champion Sabalenka and Magdalena Frech, will play Gauff on Saturday.
"I've played them both before, they're tough opponents," Gauff said.
Third seed Jasmine Paolini plays fifth seed Zheng Qinwen in another quarter-final later in the day, while Ekaterina Alexandrova faces Wang Xinyu.