The 26-year-old Baghdatis recovered from a set down to reach his fourth quarterfinal in 2012.
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The 20-year-old will face either Marcos Baghdatis or second seed Rafael Nadal in the third round.
"It was the best win of my career, " Baghdatis told the official ATP Tour Web site.
Cypriot Marcos Baghdatis eased past Frenchman Richard Gasquet, the fourth seed, with a 6-4 6-4 win.
Watching the outburst, I wondered how Tecnifibre, the racket manufacturer that sponsors Baghdatis, felt about his display.
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The Argentine, 28, broke Baghdatis four times in his first seven service games on his way to winning the title.
The 23-year-old Scot, still yet to drop a set at the event, next plays Jurgen Melzer who beat Marcos Baghdatis.
With Cypriot eighth seed Baghdatis landing only 39% of his first serves in the opening set, Nalbandian took it with ease.
That counted for nothing, though, as Baghdatis's from fell away dramatically and four successive errors handed Murray a break at 3-2.
The Croatian world No. 11 will next play fifth seed Marcos Baghdatis of Cyprus, who beat 2007 champion Philipp Kohlschreiber 6-3 6-4.
Murray will meet Marcos Baghdatis in the third round after his Cypriot opponent Grigor Dimitrov retired through injury when trailing 7-5 4-1.
Second seed Marcos Baghdatis suffered a 6-3 6-4 defeat by Robin Haase.
The fourth seed beat American Tim Smyczek 6-0 7-5 4-6 6-3 and will now play 2006 Australian Open runner-up Marcos Baghdatis .
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France's Jo-Wilfried Tsonga will meet Tomas Berdych, from the Czech Republic, in the last eight after he beat Marcos Baghdatis 6-2 7-6.
The Argentine won 6-3, 6-4, and will now face unseeded Bulgarian Grigor Dimitrov who came from a set down to beat Marcos Baghdatis.
Argentine Nalbandian, a former SW19 finalist, lost in straight sets to Marcos Baghdatis while Safin went down meekly to eventual champion Roger Federer.
As the fourth seed he receives a bye through to the second round, at which point he will meet Marcos Baghdatis or Sam Querrey.
Suddenly, Baghdatis was right back in the match with a love-set and a break at the start of the fourth after a Murray double-fault.
In other first round matches, Ukraine's Alexandr Dolgopolov beat Marcos Baghdatis of Cyprus 6-4 6-4 and Spain's Nicolas Almagro defeated Poland's Lukasz Kubot 7-5 6-2.
Meanwhile second seed Marcos Baghdatis has been forced to pull out of the tournament because of a shoulder injury he suffered in practice on Sunday.
Baghdatis continued his hot streak in Australia to dump Ferrer out in an enthralling match that lasted four hours and nine minutes on Hisense Arena.
Also advancing were No. 22 Fernando Verdasco, No. 28 Marcos Baghdatis, the 2006 Australian Open runner-up, No. 31 Radek Stepanek and No. 32 Julien Benneteau.
The World No.3 will play Grigor Dimitrov of Bulgaria who reached his first ATP final by edging out Marcos Baghdatis of Cyprus 6-3 5-7 7-6.
"He's a pain when he plays good, " Baghdatis said of Nalbandian.
He was joined by the Czech Republic's Tomas Berdych after the second seed's opponent, Marcos Baghdatis, retired hurt with a foot injury in the first set.
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The other semifinal will be between 2006 Australian Open finalist Marcos Baghdatis of Cyprus and 21-year-old Bulgarian Grigor Dimitrov, who upset Austrian seventh seed Jurgen Melzer.
But the 2002 champion clawed the break back and then struck again for a 6-4 6-4 4-6 7-5 win and a last eight match with Marcos Baghdatis.
Simon, who reached the semifinals in Brisbane last year, will play Marcos Baghdatis in the next round after the Cypriot defeated sixth-seeded German Florian Mayer 6-4 6-2.
Haas will be playing his first quarterfinal of the year against Cypriot Marcos Baghdatis, who beat qualifier Dustin Brown 1-6 6-4 6-1 in one hour and 47 minutes.
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After requiring a fairly tortuous rain-hit two days before seeing off Juan Ignacio Chela on Thursday evening, Murray was back on court 20 hours later to face Baghdatis.
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