Spending almost three hours on court, Vekic had to win the last four games of the match to secure her 7-6 (7-4) 6-7 (7-3) 7-5 victory on Margaret Court Arena.
Down 5-3, the Croatian rallied to make the round of 16 at Melbourne Park for the third time.
Vekic had the majority of the crowd behind her, something she felt gave her an edge over 20-year-old Shnaider, who is coming off a breakout 2024
"It's not without a reason that this is one of my favourite courts, so many Croatians in the crowd," the 18th seed said.
"It was tough, she (Shnaider) was playing unbelievable.
"In my next life I want to be a lefty, I can't remember the last time I beat a lefty."
Vekic, whose best result at the Australian Open is the quarter-finals in 2023, will next face veteran Anastasia Pavlyuchenkova.
Vekic VICTORY 💙— #AusOpen (@AustralianOpen) January 17, 2025Donna Vekic overcomes Shnaider in an (almost) 3-hour showdown 7-6(4) 6-7(3) 7-5 to advance to the fourth round!@wwos • @espn • @eurosport • @wowowtennis • #AusOpen • #AO2025 pic.twitter.com/GBATuDLzke
The Russian 27th seed produced a 6-1 6-2 demolition of Germany's Laura Siegemund, who had eliminated last year's Australian Open finalist Qinwen Zheng in the second round.
"The three girls I beat here, I lost to them (last year), so let's keep it going," Vekic said.
The winner of the Vekic-Pavlyuchenkova match could meet dual defending champion Aryna Sabalenka in the quarter-finals.
After winning last year's Open without dropping a set, Sabalenka has extended that streak at Melbourne Park in winning her first three matches.
Spanish 11th seed Paula Badosa has equalled her best result at an Australian Open after defeating Ukrainian Marta Kostyuk 6-4 4-6 6-3.
Badosa took two hours and 32 minutes to take down world No.18 Kostyuk, who reached the quarter-finals last year at Melbourne Park.
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