The 34-year-old Scot, who was granted a wildcard for the first Grand Slam of the year, prevailed 6-1 3-6 6-4 6-7 6-4 in just under four hours on the John Cain Arena.
World No 113 Murray played Basilashvili last week at the Sydney Tennis Classic on his way to the final and defeated him in three tight sets and it was another close encounter against the Georgian, who is seeded 21st for the tournament.
Three years ago, it appeared Murray was on the brink of retirement after an emotional farewell following a five-set defeat by Roberto Bautista Agut, but following hip resurfacing surgery, the former finalist is back at Melbourne Park.
The match against Basilashvili was a rollercoaster, with breaks of serve and sets traded, but Murray’s defence nous brought enough errors from his opponent to get the win.
"It’s amazing to be back,” Murray said. “It’s been a tough three or four years. I’ve put in a lot of work to get back here. I’ve played on this court many times and the atmosphere is incredible.
"This is the court I thought potentially I’d play my last match three years ago, so it’s amazing to be back and winning a five-set battle like that. I couldn’t ask for any more. I’m so happy.”
Murray will now face Japan’s Taro Daniel, ranked 120 in the world, in the second round on Thursday.