The second seeded Russian had a meltdown as the heat was turned up and had to dig deep to save two match points before staging an epic comeback to beat courageous compatriot Alexandra Panova 6-1, 4-6, 7-5.
British sixth seed Murray had a much easier time, cruising into the third round with a 6-1, 6-3, 6-2 drubbing of local hope Marinko Matosevic.
Other seeds progressing included men's seventh seed Tomas Berdych, 24th seed Richard Gasquet and women's 10th seed Ekaterina Makarova.
While the searing 42 Celsius (107.6 Fahrenheit) heat of last year has yet to materialise in Melbourne, temperatures reached 32 C and players, including Sharapova and Murray, draped ice towels around their shoulders at the changeovers.
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At the 2014 tournament extreme heat forced organisers to suspend play for several hours following days of complaints about "inhumane" conditions which left some players fainting and vomiting.
Sharapova, who can unseat Serena Williams as world number one if she wins the title, wilted badly after winning the opening set, firing a slew of unforced errors that almost saw her on an early plane home.
"I was one point away from being out of this tournament twice today and not playing my best tennis so I'm just happy I was able to win that last point.