The 25-year-old Briton, who captured a long-awaited major title at the US Open last year, charged out of the blocks against the 53rd-ranked Haase, capturing an early break in each set and wrapping up the match in a quick-fire 97 minutes.
Haase had previously proved a handful for Murray, taking him to five sets in the 2011 US Open and beating the Scot in their only other match at Rotterdam in 2008.
But the 25-year-old Dutchman never threatened in the sunshine at Rod Laver Arena.
Murray said that playing as a Grand Slam champion did not feel any different.
"Not really, it didn't feel much difference to me. I was still nervous before I went on to the play the match," he said.
"I think when I would see the benefits of it is if I get myself deep into a Slam this year and you're playing against the top players.
"That's when I think you'll draw on that experience and use it in the right way."
Murray, twice a finalist at Melbourne Park, will next play Portugal's Joao Sousa, who beat Australian wildcard John-Patrick Smith.
The match was the first on Rod Laver Arena and Murray set about his business quickly with two breaks of serve to move 4-1 up.
He survived five break points in holding the following game before Haase then held and took one break back to stay in the set, but Murray broke again to seal the set.
The second set took only 26 minutes and Haase looked a beaten man when the Scot broke to love at the start of the third.
Haase did break back, but Murray would break him for an eighth time - Haase blasting a forehand long - to finish the job.
Laura Robson reached the second round of the Australian Open for the first time with a powerful 6-2 6-3 win over Melanie Oudin.
The British number two, who will turn 19 next week, joins Heather Watson in the next round after the British No.1 won on the opening day.
Robson will face eighth seed and former Wimbledon champion Petra Kvitova in round two.
She captured the imagination of the British public with a run to the last 16 of the US Open in 2012 - and will take a similar mentality into her next clash.
"I'll go in with nothing to lose again," said Robson. "I seem to play pretty well when I do that.
"It's going to be tough, two lefties playing each other. I haven't played a lefty in a while, so I'm going to have to try as best I can."
Oudin, 21, is a former starlet herself but is yet to live up to the hype generated when she reached the fourth round of both Wimbledon and the US Open as a 17-year-old in 2009, beating Maria Sharapova at Flushing Meadows in the process.
Robson, ranked 53rd in the world to Oudin's 84th, had few problems in the heat of the afternoon.
She won the first four games in claiming a quick first set then went a break up early in the second.
Oudin broke back thanks to a pair of double faults but Robson powered her way to the set as the American's forehand let her down.
"I think I played pretty well, just not as well as I would have liked to. Hopefully in the next match it will be a bit better," added Robson.
"I can hit it pretty hard these days but I think the key is moving around the ball enough to get it to where I want it to go every time. Today I just hit a few balls late here and there, where I should have just moved my feet a little bit."
Glaswegian Jamie Baker, meanwhile, went out of the men's draw at the first hurdle after losing in straight sets to Czech Lukas Rosol - the man who knocked Rafael Nadal out at Wimbledon last year.
World number 246 Baker served for each of the first two sets but was broken on each occasion as he went down 7-6(5) 7-5 6-2.
Rosol is ranked 75th in the world.
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