He will be joined in the last 16 by Daniil Medvedev, who overcame tenacious Italian Matteo Arnaldi 5-7, 6-4, 6-4, and Carlos Alcaraz after his chief rival beat China's Wu Yibing in straight sets.
A weary-looking Sinner dropped the first set on a tiebreak to the 37th-ranked Etcheverry, just three days after losing a 3hr 21min China Open final in Beijing to Alcaraz.
But the 23-year-old recovered in the second set, breaking Etcheverry twice to level the match before easing to victory.
"Tomorrow is one day off which I really need, I felt it physically today," Sinner said.
"I had some chances in first set but couldn't use them... I'm very happy with how I bounced back," he said.
World number five Medvedev broke early in the first set but 36th-ranked Arnaldi kept his cool to break back in the sixth game on an unforced error from the Russian.
Arnaldi broke again in the 12th game after Medvedev hit long.
The Russian's mood darkened under pressure from Arnaldi in the second set and he was slapped with two code violations and a point penalty after arguing with the umpire.
The 28-year-old former world number one recovered to take the match to a deciding set and came through after 2hr 44min.
"I was expecting him to play differently, he played very good," Medvedev said of Arnaldi.
"To be honest in the first set I should have done better. The second and third were even closer... but at the end I'm happy to win."
Alcaraz wins again
World number two Alcaraz had a slightly easier time, defeating Wu 7-6 (7/5), 6-3.
Wildcard Wu did well to hold off the Spaniard until the tiebreak in the first set.
Four-time Grand Slam winner Alcaraz hit his stride in the second set, breaking in the sixth game.
Wu was ranked at a career-high 54 last year but injury has seen him slip to 560th.
"Once again he's shown his level, he deserves to be in the top (100)," Alcaraz said of Wu, to roars of approval from the home crowd.
"I'm very happy to get through, it was a tough match."
Wu is the second Chinese player Alcaraz has defeated this tournament, having beaten Shang Juncheng on Saturday.
"Thank god I'm not going to play a Chinese player again, I hope more people will cheer for me!" Alcaraz joked.
Play was only possible under the roof of the main court, with rain forcing matches on outside courts to be postponed for the second consecutive day.
