The Portuguese was brushed aside 6-2 6-2 6-4 as the world number three advanced to a last-32 meeting with Ricardas Berankis.
Importantly, on the hottest day of the tournament so far - temperatures moved beyond 100F in Melbourne - Murray got the job done in just an hour and 41 minutes, conserving energy for what he hopes will be a concerted push for a second successive grand slam crown following his maiden major win at the US Open.
"It doesn't matter how much training you do, it's tough in these conditions," he said.
"It's extremely hot, especially when the sun comes through the clouds. It's good to get it done in three sets.
"The longer the rallies go the tougher it becomes on the legs.
"You need to get in control of the points and dictate them because otherwise you will get very tired very quickly."
The match itself was a no-contest, the outcome never in doubt with Sousa's game simply not big enough to cause Murray any significant problems.
The world number 100 had limited chances to impress as Murray dictated the match from the baseline.
And on the rare occasion he had to escape from trouble, the Scot's serve came to his aid.
Murray broke to establish a 2-1 lead and although Sousa threatened an immediate response at deuce in the next game a wonderful backhand pass up the line and an ace saw the Scot extend his advantage.
Another break put Murray in total control and he had few alarms in serving it out to move a set up.
The second set continued in the same vein with Sousa unable to make any inroads.
A horribly shanked forehand handed Murray another break in the first game, a gift he hardly needed.
And when Murray claimed the Sousa serve once more, having chased down a drop shot and forced an error from his opponent, for 3-0 the writing was on the wall for the 23-year-old from Guimaraes.
To his credit, he got on the board in the fifth game - albeit with plenty of help as Murray's concentration wavered - but it did little to stem the tide as Murray held for 5-1 having been taken to deuce.
Two holds followed as Murray cruised into a two-set lead.
Any remote hope the crowd had of a Sousa comeback was evaporating by the minute and Murray kept his foot on the gas by starting the third set with yet another break - his fifth of the contest.
A powerful forehand return put him 30-40 up and, having moved his opponent round the court, he took the chance when Sousa could only put a desperate lob beyond the baseline.
Murray was clearly in the mood for a quick finish and he showed his desperation to get the job done with a loud yell of anguish after putting a backhand into the net in the seventh game.
He was irritated further as a break point came and went before Sousa, via a neat serve and volley, took it to remain just one break down at 4-3.
It was to prove enough though as two confident holds saw Murray progress to a meeting with Berankis.
"I have practised with Berankis a lot," said Murray.
"He is a very hard worker, he has had a couple of injuries over the last couple of years and it will be tough."