The British third seed looked intent on keeping his time in the heat and humidity to the absolute minimum, wrapping up victory in just 101 minutes when his Portuguese opponent went long with a return.
Two breaks in the first set, a couple in the second and one more in the third proved enough to set up a meeting with Lithuanian Ricardas Berankis, who earlier stunned German 25th seed Florian Mayer 6-2 6-3 6-1.
The convincing natuure of Berankis's win over Mayer indicated that he might prove trickier for the Scot than the game but limited Sousa.
Murray unveiled his full repertoire of shots in his contest and every one was superior to that possessed by his Portuguese opponent.
The 25-year-old stuttered a little as he looked to close out the match in the third set but fired down his 14th ace to bring up match point before clinching the victory when a Sousa return went long.
"It doesn't matter how much training you do, it always difficult when the sun comes out here," Murray said.
"You need to get in control of the points and dictate them because otherwise you will get very tired very quickly in these conditions."
Murray, who is chasing a second successive Grand Slam title after his breakthrough triumph at the US Open, has reached the semi-finals at Melbourne Park for the last three years.
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- Florian Mayer