The number 14 seed also overcame Victor Troicki in four sets in his second-round clash and he took the opener, but the Georgian transformed in the second to level after consolidating an early break of serve.
'It's a stunner!' - Basilashvili fires fierce backhand past advancing Tsitsipas
Basilashvili recovered from 5-3 down to force a tiebreak in the third, but Tstisipas put a bizarre call from the umpire and line judge behind him to edge it 9-7.
The talented 20-year-old, roared on again by a strong Greek contingent at Melbourne Park, took a crucial break to move 4-3 up in the fourth, and he twice then held serve to confirm his progress.
Tstisipas will now face Roger Federer for a place in the quarter-finals.
Highlights: Tsitsipas beats Bashilashvili for place in last 16
"It was tough and he pushed me a lot," he told Eurosport. "I was missing a bit more than usual, and his ball speed was a lot different to the rest of the players this week. I managed to stay patient and stay in the rallies.
"Mentally it was tough at times to deal with it. The conditions were a bit faster than usual. I was struggling at first with my serve but there was no sky - and things like that matter, even though it sounds stupid. But it worked out well once I adjusted.
Tsitsipas: It was mentally tough at times
On the controversial line judge call, he added: "I served a good serve wide, and to me it felt like the set was over. Suddenly, the umpire said the linesman had called it out. It was very late.
"It was very frustrating and it was one of those moments when it felt like they'd taken something away from me."