The Briton, 30, said he is optimistic about his future and is aiming to play at Wimbledon this July following the operation on his right hip by Dr John O'Donnell, one of the world's leading hip surgeons.
"Today I underwent successful right hip surgery at the St Vincent Hospital in Melbourne," said the three-time grand slam champion on his Facebook
"I look forward to returning to competitive tennis during the grass court season. Thanks to everyone for all the well wishes and support over the last few days. I'll come back from this."
Murray, now ranked 19th in the world, has not played competitively since July 2017. The Scot attempted a return at the US Open in August but pulled out two days before the start of the tournament and last week he withdrew from the Australian Open, which starts on January 15.
The two-time Wimbledon champion wrote on Instagram last week that surgery was his "secondary option" because "the chances of a successful outcome are not as I high as I would like."
His reluctance to be operated on was understandable. Players who have had hip surgery in the past, such as former world No.1s Gustavo Kuerten and Lleyton Hewitt, have struggled to recapture their previous form.
Two-time grand slam champion Hewitt failed to reach the latter stages of grand slams after undergoing surgery in 2008, while Kuerten's best after going under the knife in 2002 was the Australian Open third round.
But Murray, speaking from his hospital bed to British tennis reporters, said he was "not finished playing tennis yet."
"I'm going to be competing at the highest level again," he added, revealing he has been in contact with O'Donnell since 2008, when his hip first started bothering him.
"I'm very optimistic about the future - the surgeon is very happy about how it went."
The Briton said he should be fit enough to fly back to Britain within a fortnight and has been given a guideline of 14 weeks for a return.
"My plan is to be back playing around the grasscourt season -- potentially before then. I'm going to take my time to make sure that the rehab is done properly, and make sure that the surgery is as successful as it can be."
By this season's Wimbledon, Murray will have been sidelined for almost a year, having last played on 12 July 2017 when he was knocked out of Wimbledon in the quarterfinals by Sam Querrey.
The father-of-two told reporters that the desire for his oldest daughter, who will be two next month, to watch him play was his motivation to return to the pinnacle of his sport.
"One of the things that I would like to do is play until my eldest daughter is able to watch me and have a small understanding of what it is I've done for my living," he said, according to The Independent.
"That's something I really wanted to do. That's one of the things that's motivated me to keep playing."