Greek ninth seed Tsitsipas, who was runner-up to Novak Djokovic at Roland Garros in 2021, won 3-6, 7-6 (7/4), 6-2, 6-2.
His reward is a clash with Carlos Alcaraz for a place in the semi-finals.
World number 35 Arnaldi from Italy had stunned sixth-ranked Andrey Rublev in the third round but wasted a golden opportunity against Tsitsipas when he let four set points in the second set slip away.
"It was one of the craziest comebacks I've had," said Tsitsipas.
"The momentum seemed to be going his way the entire match, it was very frustrating on my end. Nothing really seemed like it was working.
"I think the spirit was the reason I managed to come back today. That game at 5-3 when I broke was the biggest pleasure I've experienced in tennis for a long time."
Alcaraz boasts a 5-0 career head-to-head record over Tsitsipas with three wins on clay.
That includes a straight-sets win in the quarter-finals in 2023.
"He has said in the past he likes playing against me, so I hope he gets to like it a little bit less this time," said the Greek.
