The 373rd-ranked Jones matched the 29th-ranked Melzer shot for shot in a first set that contained no breaks of serve. He rallied from 6-4 down in the tiebreaker before closing out the second set in only 30 minutes, belying a ranking that had slumped from 187th.
"It's a pretty good feeling," said Jones, who won three matches in qualifying. "I had a pretty rough year last year, which my ranking shows. I trained really hard in Melbourne with a couple of guys, a couple of fellow Aussies during the offseason, and physically I'm just feeling a lot better on the court."
Alejandro Falla of Colombia was leading Grega Zemlja 6-4, 3-1 when the Slovak retired with an unspecified injury, while Canadian qualifier Jesse Levine beat New Zealand wild card Dan King-Turner 6-2, 6-2.
Gael Monfils rallied past Benjamin Becker of Germany 6-7 (2), 6-3, 6-4. The Frenchman, ranked 99th, entered the main draw on a wild card and looked in good form ahead of the Australian Open, which starts Jan. 14.
Yen-hsun Lu of Taiwan beat Benoit Paire 6-3, 2-6, 6-2 with the Frenchman appearing to struggle with a hamstring injury.