The 23-year-old, ranked 185 and in his main draw debut at Melbourne Park, blasted past the Italian 7-6(1), 6-4, 6-1 on John Cain Arena.
"Very satisfying. I was already happy to qualify here for my first time, and now obviously coming through qualies and winning a round, I'm proud of it," he said.
Fery's only other Grand Slam second round appearance came at Wimbledon last year, where he also beat a 20th seed in the first round - Australian Alexei Popyrin.
When he successfully negotiated qualifying in Melbourne his mum Olivia, who played in the 1991 Roland Garros women's doubles, was quick to jump on a plane to Australia and support him.
"When I won that last round (of qualifying), she asked me if she could come," he said.
"I was, 'Of course, it would be nice to have you here'. It's obviously a long way, but at least I made it worth it. She can see at least two matches - hopefully more."
Asked if she had given him any tips, given her own professional tennis career, he replied: "When I was younger. Now, not so much.
"She leaves me with my coaches to talk about tennis. But like everyone's mum and dad I guess they give you life advice."
Defeat was a big setback for Cobolli, who appeared to be battling a stomach issue throughout the match.
He made the Wimbledon quarter-finals last year, before falling to Novak Djokovic, and won two ATP titles at Bucharest and Hamburg.