Trailing two sets to one when the tie was halted the previous evening, Tsitsipas came through 7-6(3),(2)6-7, 4-6, 7-6(3), 6-4.
After a gruelling four-hour 40-minute match-up, the Greek goes on to face Laslo Djere of Serbia for a place in the last 16.
For Murray, the defeat meant he still has not made the second week of a Grand Slam since reaching the quarter-finals at Wimbledon in 2017.
It will also surely raise questions over his long-term future in the sport.
"It's never easy against Andy, everyone here loves him," said Tsitsipas, who hit a whopping 90 winners.
"I am always impressed by him after so many years on tour and to keep playing after two hip surgeries. I wish him all the very best for the future."
Murray had been ahead when the match was halted on Thursday due to a local 11:00 pm tournament curfew.
The suspension came at the right moment for the former world number one, who fell and appeared to hurt his groin, leaving him screaming in pain as he served for the third set.
However, the world number 40, who had secured the only break of serve in the first three sets, picked himself up to serve it out.
Murray showed no sign of injury after his dramatic fall and was given raucous support again from the stands.
But Tsitsipas was incredibly composed to edge a tense fourth set on a tiebreak after both players had been rock solid on serve, as they had been throughout the contest.
Murray may rue not challenging a line call that went against him at 4-4 in the fourth set when leading 15-30 on the Tsitsipas serve. His return was called out, he did not challenge, and the ball was shown to be in.
"I mean, it was right underneath the umpire's nose. They shouldn't be missing that, to be honest. If they're unsure, they should let the player know," Murray said.
"You can obviously argue it's my mistake. Ultimately the umpire made a poor call that's right in front of her."
With a 12-year age difference, Tsitsipas suddenly found another gear and broke for the first time in the match to take a 2-1 lead in the decider, claiming victory on a third match point.