Eighth seed Berrettini, who has dropped only one set in the tournament, will be aiming to win his maiden ATP Masters 1000 title against Zverev on Sunday.
It was a patchy performance from the world number one, who is nursing an abdominal injury and drew deeply on his mental fortitude to win 6-7 (6/8), 6-2, 6-4, 7-6 (8/6).
The surprise of the men's draw has been Russian qualifier Aslan Karatsev, who will be looking for a semi-final place in his first Grand Slam tournament when he faces Bulgaria's 18th seed Grigor Dimitrov.
The US Open finalist got a high percentage of first services in and never appeared to lose control against his fellow 23-year-old during the final match of the evening session on the Rod Laver Arena.
Zverev hit six more winners than Cressy, who is ranked 172nd.
"It's tough to say because it can both be negative and positive," said the Russian, whose best Slam performance so far was making the final of the 2019 US Open.
Top seed Wawrinka, who sent down 14 aces and converted four out of seven breakpoint chances in the match, will continue his bid for a first ATP crown since winning in Geneva in 2017 and takes on local favourite Alex Bolt on Thursday.
After a year of playing in front of mostly empty stadiums due to Covid-19, he is also looking forward to some atmosphere, with up to 30,000 spectators allowed each day at the Australian Open.