Bidding for his maiden title at Melbourne Park after three trips to the final, the sixth seed had few problems in carving out a 6-1 6-3 6-2 win in one hour and 45 minutes on a steamy day on Margaret Court Arena.
The last Briton standing following the exit of Heather Watson, James Ward and Kyle Edmund on Tuesday, he will next face Portuguese No 1 Joao Sousa in the third round.
Murray raced out of the blocks and wrapped up the opening set in just 21 minutes but had some tighter games in the second and third sets.
The Scot broke the erratic Matosevic's serve seven times and controlled the match, with the Australian committing 49 unforced errors to Murray's 12.
Matosevic's first-round opponent, Alexander Kudryavtsev, accused boisterous Australian fans of behaving like "animals" after losing in five sets to the Melbourne resident.
Murray, who is good friends with Matosevic, beat the Australian in the second round of last year's French Open and at this year's Hopman Cup, both times in straight sets.
Murray improved to 10-0 against players from Australia with the win and advanced to meet Sousa, who advanced after Slovakian Martin Klizan retired in the fourth set of their second-round encounter.
After the match, Murray said of his record: "I love Aussies. I get on well with a lot of the players from Australia and it's just luck – I had no idea that was the case.
"But I'm sure in the next few years with the players the Aussies have got coming through, some fantastic young players, I am not sure if I am going to hold on to that record for much longer."
Matosevic, who had lost all 12 of his Grand Slam matches, won just three points in the first five games and received a sympathetic cheer from the crowd when he finally troubled the scorers to reduce his arrears.
However, Murray looked in no mood to share such sympathy and closed out the set with ease to leave Matosevic with a mountain to climb to become the first Australian to beat Murray at tour level.
A double fault from Matosevic helped give Murray a break of serve in the opening game of the second set, but it was two moments of brilliance from the Scot which secured another break in game five.
Matosevic could only look on as a backhand return from Murray sailed by for a winner and then was unable to retrieve a perfectly judged lob that landed just inside the baseline.
An apparent lapse in concentration from Murray allowed Matosevic to instantly get one of those breaks back and it was with some relief that the former Wimbledon champion held his next service game to lead 5-3.
And Murray then pressured Matosevic into more unforced errors to break serve again and take the second set, with just over an hour on the clock.
It became just a matter of how quickly Murray could seal his place in the third round and he recovered from 40-0 on Matosevic's serve to break for a 2-0 lead.
The three-time Australian Open finalist had to save a break point to consolidate his advantage – much to his annoyance – as the home fans tried desperately to inspire Matosevic to make it more of a contest.
However, Matosevic was soon facing match points on his own serve and although he saved the first two, two forehand errors completed a miserable performance and gifted Murray another break to complete a comfortable win.