Murray, who enjoyed a straight sets win over Viktor Durasovic in the earlier round, was made to work a lot harder against Basilashvili who won the opening set.
Despite the early disappointment, the 34-year-old battled his way back into the game and won 6-7 (4) 7-6 (3) 6-3.
Few would have expected the match to last over three hours when Murray raced to a 4-1 lead in the opening set.
However, a period of carelessness from the Scot allowed Basilashvili to make up for lost ground and win the first set via a tiebreak.
Murray rediscovered his groove in the following set, but was forced into another tiebreak as the Georgian continued to cause him problems.
The three-time Grand Slam winner eventually prevailed and enjoyed a far more comfortable final set to seal his place in the next round.
In his post-match interview, Murray said: "I do like the physicality but I'd rather make it easier on myself. Basilashvili is hitting the ball consistently bigger than anyone on tour and I was having to do a lot of defending and running - my legs held up well in the end.
"Sometimes it felt I was hitting good serves and he was hitting massive shots. But in these conditions with no wind he is very comfortable and he was hitting huge balls one after the other and I was just having to fend them off.
"Yes I got a lot of free points with my serve. The rallies were on his terms a lot of the time, so I served a bit bigger midway through the second set. When he was serving I tried to take time away."
On quarter-final opponent David Goffin, Murray said: "He had a tough year last year, missing a lot of tennis due to surgery. He's on his way back, but he's been up in the top 10, 15 in the world for many years before his injuries and I'm sure he'll want to get back there."