The Ukrainian, who scored her biggest career win at the WTA Finals in Singapore last October, triumphed in a draining 4-6, 6-4, 7-5 roller-coaster that included medical time-outs for both players.
"It was very hot conditions and I was struggling a little, we left everything on court," the 24-year-old said. "I told myself I'm gonna die or win.
"It was just immense, I had to stay very strong and we were fighting until the very last point."
The Ukrainian, who will meet either Elise Mertens or Madison Keys in the next round, went over to congratulate Zhang after the two hour 55 minute clash.
She received treatment on her neck during the match, while Zhang had problems with her knee.
Svitolina, a quarter-finalist at Melbourne Park last year, conceded the first set and was 3-0 down in the third before she rallied.
With boyfriend Gael Monfils cheering her on, she took five straight games, including three breaks of serve to take it to 5-3.
She faltered and conceded a break serving for the match but immediately broke back before claiming another two games to seal the win.