Murray required just 72 minutes to sink the world number 345 and bounce back from his disappointing loss to Facundo Bagnis in Melbourne to win 6-3 6-1.
The former world No 1 has accepted a wild card for the Australian Open, three years after intimating he may have played his last game due to impending hip surgery.
Murray said in his on-court interview: "It is really nice to get my range and hopefully I can carry that through to Melbourne.
"I was hoping to get matches in Melbourne last week, but that didn't happen. Thankfully the tournament organisers agreed to give me the wild card and the opportunity to play here and I am very grateful for that."
Murray produced a strong performance from the baseline to outclass Durasovic, not losing his serve once and breaking the Norwegian four times. The Scot will next face Georgian second seed Nikoloz Basilashvili.
That match is due to be played at 9.30am GMT on Wednesday. It will be the second time in Murray’s career that he has faced Basilashvili, with the Scot defeating him in the first round of Wimbledon last year under the roof on Centre Court.
British No 2 Dan Evans is also in the Syndey field and he takes on Spain’s Pedro Martinez in the early hours of Wednesday morning. Top seed Aslan Karatsev of Russia opens his account against Serbia’s Miomir Kecmanovic.