This was a much stiffer examination than his first-round win against Viktor Durasovic on Tuesday, with the three-time grand slam winner progressing past the number two seed 6-7 (4) 7-6 (3) 6-3 after over three hours on court.
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."
Murray joins fellow Briton Dan Evans in the quarter-finals after he cruised to a straight sets victory over Spain's Pedro Martinez.
The British third seed was a class above against his 61-ranked opponent, dominating on both sides of the court en route to a 6-2 6-3 victory.
Evans converted all four of his break point opportunities and only gave up eight unforced errors as he continued his strong run of form.
He will face the winner of sixth seed Dusan Lajovic and French-American Maxime Cressy's match.