Ranked 153 places below the world No.45, Barton went game for game with Edmund for the entire match but earned his 7-6 (7-3) 7-6 (7-5) win with timely points in both tiebreakers.
The Sydney-born Barton last year beat then-world No.32 Steve Johnson in Auckland and also progressed to the second round of his grand slam debut at Wimbledon.
The 25-year-old rated the win in his hometown debut better than the one in Auckland.
"Even though that (Auckland) was my first tour event win, it was special as well," Barton told AAP.
"But to do it here in front of all my mates and family and everyone supporting me, it's just unbelievable.
"He was serving unreal and I was serving well. A few break points here and there but it was real tight to get into any services games.
"I just knew I had to take care of my own service and I hoped to jag a few points in the tiebreak, which I did. I was lucky enough in the end to get balls going my way."