Momentum swung both ways with Basilashvili giving the match his all before Murray won with 6-1 3-6 6-4 6-7 (5-7) 6-4.
Murray was one of the four Brits who were successful on Tuesday. US Open Champion Emma Raducanu, Dave Evans, and Heather Watson also progressed further.
Murray is set to face Taro Daniel on Thursday. In his interview, Murray said, “It’s been a tough three or four years but I’ve put in a lot of hard work to get here.
“I’ve played on this court many times and the atmosphere is always incredible.
“It is where I thought I had potentially played my last match but to win a five-set battle like that, I couldn’t have asked any more.”
Three years ago, Murrays career was almost sidelined with a hip injury that left him in chronic pain. He underwent a resurfacing surgery and had metal inserted into the joint. Murray is the first player to have successfully returned to professionally playing after such a surgery.
Currently ranked 113th in the world, Murray is hoping to start playing in major tournaments again. When asked about his aims for Melbourne, he said, Hopefully keep improving - there are things in my game I could do better.
“I would like to have a deep run. It’s something I haven’t had since I came back from injury - and I hope I can do it in this tournament.”