At about 12.30am overnight, police were called after alarms were activated on a number of business premises on Henley Beach Road at Torrensville.
Officers allegedly found several businesses had been broken into, along with a 32-year-old man with an injured leg.
Police claim the man hurt his leg as a result of the break-ins.
After speaking with the 32-year-old, police attended a nearby home in Cowandilla where they found the body of a 60-year-old man.
Police believe the 60-year-old was a resident at the home, and that he and the 32-year-old were known to each other.
The 32-year-old man, from Cowandilla, was arrested and is being treated for his leg injury.
It is expected he will be charged with murder and property offences once treatment is complete, police said.