Playing just his second tournament at Melbourne Park in five years due to two major hip surgeries, the Scotsman fashioned a 6-1 3-6 6-4 6-7 (5-7) 6-4 victory in a marathon clash on John Cain Arena on Tuesday.
"It has been a tough, tough three or four years and I put in a lot of work to get back here.," Murray said in his on court interview.
"I played on this court many times ... this is the one where I thought potentially I had played my last match three years ago, but it is amazing to be back winning a five- set battle like that.
"I couldn't ask for any more."
The 34-year-old looked like he may run out of steam heading into the fifth set, having already played five times last week en route to the final in Sydney.
Granted a wildcard into the Open, Murray who made five finals between 2010 and 2016, dug deep and broke the Georgian early in the deciding set, taking a 4-1 lead.
But Basilashvili, who he beat in the second round in Sydney, wouldn't bow down and clawed his way back to level at 4-4.
Basilashvili's serve then failed him at the final hurdle, giving Murray three match points to secure the win.
"It was a brutal match in Sydney, we played three hours over three sets," Murray said.
"He hits the ball huge from the back of the court and I had to do a lot of running today, a lot of defending.
"He is not the sort of guy whose name you want to see next to yours in the draw very often so I took obviously a bit of confidence from the win last week, and thankfully I managed to get over the line today."
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