Last year's runner-up dropped his first set of the year, and was made to struggle before overcoming Ukraine's Alexandr Dolgopolov 7-5, 6-3, 6-7 (3/7), 6-3.
The British fifth seed needed 3hr 06min to book a place in the last four, where he will play Spanish seventh seed David Ferrer on Friday.
Murray, who lost in three sets to Roger Federer in last year's Australian final, improved his unbeaten run in the new season to eight matches.
"I feel just way more experienced. I know how to deal with playing deep into Grand Slam events now, how to get prepared for them mentally and physically," Murray said.
"It's something that I'm a lot better at. I've tried to become more consistent, have fewer weaknesses. I think this year I'm a little bit more solid.
"It's always very tough when you come up against those guys. You need to be on your game physically and mentally if you want to beat them. So that will be the case in the next match."
The streaky Dolgopolov proved a handful with his 57 winners, offset by 77 errors, while the steadier Murray had 33 winners and 34 errors.
It took the Scot a set to work out the quirky, 46th-ranked Ukrainian's game, before he went two sets up and was seemingly poised for a straight-sets victory.
But Dolgopolov hit back, taking advantage of Murray's sloppiness to win the third-set tiebreaker and force the match into a fourth set.
As a measure of his difficulties, Murray had lost a total 22 games in his four matches leading up to the quarters, but Dolgopolov took 18 games off him.
"He's just unorthodox, very different to how most guys play," he said.
"It's tough to get into a rhythm. But he's also a very good player, definitely not someone to be underestimated.
"I wouldn't say I was necessarily in trouble at any stage. I was ahead in most of the sets. Getting ahead early in the fourth set made a big difference.
"But I thought I dealt with his game well. It was just difficult to get into a rhythm."
Murray was broken twice in the opening set, but he broke the unpredictable Ukrainian's serve three times to go one set up in 57 minutes.
The puzzled Scot took some time to figure out the free-spirited Dolgopolov, who constantly went for his shots and ran up a stream of errors in the process.
Flashy Dolgopolov hit 21 winners in the opening set, but they were offset by 26 mistakes.
Murray grabbed a vital break in the fourth game of the second set when Dolgopolov casually over-hit a volley, when it was easier to play a routine winner on game point.
The determined Scot tightened up his serving in the second set, playing the percentages and not dropping a point on his opening four service games to take a grip on the match.
Dolgopolov played another erratic game to lose his opening service in the third set, but he broke back in the sixth.
The Ukrainian had another purple patch in the tiebreaker, as the rattled Murray made three crucial errors to drop his first set of 2011 and take the match into a fourth set.
Dolgopolov began the fourth set disastrously, dropping his serve to love, and Murray rammed home with another service break to lead 4-0.
The Scot gave one break back with a sloppy game, but recovered his consistency to get over the line.