The British hopeful, whose previous best finish at a Grand Slam came at the US Open in 2016, recovered from a mid-match slump to prevail 7-6(0) 3-6 4-6 6-0 7-5 in a time of three hours and 38 minutes.
After losing serve in the opening game of the contest, Edmund hit back in the fourth and took the set to a tie-break, which he ran away with 7-0.
With the temperature hitting 39°C in Melbourne, the world number 49 struggled to keep his composure and lost seven games on the bounce en route to falling 2-1 behind in the match.
The second game of the fourth set would prove pivotal, as Edmund broke his opponent's serve after a 20-minute tussle that included eight break points.
In the same time it took to win that one game, Edmund would take each of the next four as momentum swung back his way.
The match now level, the fifth set proved to be a far tighter affair which was edged by Edmund, taking advantage of Basilashvili's double-faults to take the decider 7-5 and set up a meeting with either Ivo Karlovic or Andreas Seppi in the last 16.