The Brit, playing as a wild-card entry, came flying out of the blocks as he took the opening set in just 23 minutes.
Basilashvili responded, breaking three times to win the second set, but Murray reclaimed the lead after taking the third set.
The Georgian gutted out the fourth-set tiebreaker, Murray, a five-time finalist, broke late in the fifth set for his third victory in the last six months over the 21st seed.
Murray edges Basilashvili in marathon
Basilashvili took the opening game of the match, but unforced errors started to creep into his game as Murray took control.
A break for a 2-1 lead was confirmed after the Georgian went long and he swept through the rest of the set, breaking to love to take it in just 23 minutes.
The 21st seed finally recorded another hold in his opening service game of the second set, punctuating it with an ace. The winners started to come and he broke Murray for the first time.
A second break followed as the five-time Australian Open runner-up dumped a forehand into the net to give Basilashvili a 4-1 advantage.
Murray rallied to close the deficit to 4-3, but Basilashvili finished strong, winning the final two games to level the match at one set apiece.
Solid serving from both players marked the beginning stages of the third set with the fifth game proving to pivotal, lasting ten minutes as the Georgian nets and Murray takes the lead.
After Basilashvili broke back, the world number 113 re-took the advantage, the 21st seed making three unforced errors in the game.
Serving for the set at 5-4, an ace on set point was called as a let, but on the replay of the point, Basilashvili goes long with a backhand to give Murray a two sets to one lead.
Breaks were exchanged midway through the fourth set as it would eventually be decided in a tiebreaker. A sublime backhand gave Basilashvili a 2-1 lead and he wrapped up the set when Murray netted to force a fifth set.
The Brit came firing out of the blocks in the decider, winning the first ten points for a 3-0 edge. The Georgian rallied to eventually level the set at 4-4, but Murray regrouped and one final Basilashvili error into the net on match point sealed a famous win.