The Ukrainian star beat Czech Marketa Vondrousova 4-6, 6-4, 6-4 in three sets.
Her determination paid off in the end.
"Not every day you play your best game, you just have to stay strong and stay calm and fight for every ball," Svitolina said post her victory.
Venus, Pliskova in action next
Veteran Venus Williams will take on Angelique Kerber in the quarters on Friday 5:30 AM IST. Karolina Pliskova will face Belinda Bencic in the remaining quarter-final draw later tonight at 11:30 PM IST.