Murray wins Queen's opener in 1,000th match of career
Murray, 37, a record five-time champion at the London grass-court tournament, outlasted the 48th-ranked Australian 6-3, 3-6, 6-3 in the 1,000th Tour-level match of his career.
The British former world number one dropped the second set and appeared to be struggling physically, grimacing after every serve.
But he hit back in the third, a vintage cross-court winner and fist pump lifting the vocal crowd as he brought up break point before dispatching it with a clinical backhand on his way to victory in one hour and 51 minutes.
Murray, set to retire either after Wimbledon or the Olympics, said: "I've not got too many wins this year, it's been a difficult season, and I did well to come through in the end.
"Any win you can get is important, that's why we play. With Wimbledon a few weeks away I want to get some matches in my legs and some confidence. It's great to get another win here."
The three-time Grand Slam champion, who competes with a metal hip, has struggled with an ankle injury and a back issue this year.
The Scot, who has plummeted to 129th in the rankings, said: "I've felt better, but it was enough today.
"My mum told me it was my 1,000th match and that's a lot of matches, a lot of wear and tear on the body and it's not easy, but I managed to push through it."