Delbonis took advantage of a series of Murray errors to run off the last five points of the decider to end the two hour, 46 minute struggle in Stadium 1.
"Thanks to everybody here for such an amazing match. It was a very intense match , very passionate, I think," the Argentine said.
Delbonis, who reached the quarter-finals this year in Rio and Sao Paulo, varied his strategy against Murray and took advantage of some weakness in the Scot's serve.
Murray failed to register an ace and won less than 50 percent of his second serves.
"I know it was a challenge," said Delbonis. "I have to mix (my shots) all the time.
"In the end, I can do my job and this is what I get."
It was a short appearance in only the second tournament of the year for Murray, who lost in the final of the Australian Open to world number one Novak Djokovic.
Delbonis will meet Frenchman Gael Monfils in the fourth round after the 13th seed dispatched Spain’s Albert Ramos-Vinolas 6-1 6-3.
In another day match, 12th seed Milos Raonic of Canada advanced to the fourth round 6-3 3-0 when Australian Bernard Tomic retired in the second set.
Tomic had complained about an injured wrist before the match against the hard-serving Raonic, who will meet Czech Tomas Berdych in the fourth round. The sixth seed beat Croatian Borna Coric 6-1 7-6(3).
Eighth seed Richard Gasquet of France advanced past Ukraine’s Alexandr Dolgopolov 2-6 6-2 6-1. His fourth-round opponent will be Croatian Marin Cilic, who defeated Argentine Leonardo Mayer 6-4 6-3.
Belgian David Goffin defeated Argentine Guido Pella 4-6 6-3 6-2 and will next meet Stan Wawrinka after the Swiss third seed beat Russian Andrey Kuznetsov 6-4 7-6(5).
(Reporting by Larry Fine; Editing by Peter Rutherford)
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