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The French top seed looked to be in trouble as Belgian Van Uytvanck eased to the first set, before Garcia found her feet to come back and overwhelm her opponent 2-6 6-0 6-1.
She will face Jasmine Paolini in the last eight after the Italian took just over an hour to dispose of Erika Andreeva 6-2 6-3.
Second seed Zhang Shuai is out though after falling to another Belgian in Maryna Zanevska 6-2 6-0.
Fifth seed Anastasia Potapova will face Zanevska next after her 6-4 4-6 6-3 win over Clara Burel, while seventh seed Danka Kovinic is also through after beating Anna Bondar 4-6 6-2 6-3.
At the Thailand Open, top seed Bianca Andreescu had no problems against Anastasia Zakharova, easing through to the quarter-finals 6-2 7-5.
The Canadian will face Marta Kostyuk next after the Ukrainian fifth seed defeated Nao Hibino 6-3 7-6 (9-7).
Sixth seed Tatjana Maria also progressed 6-0 6-4 against Dayana Yastremska, as did Heather Watson as she fought hard to see off Han Na-lae 6-3 5-7 6-4.