In the third set, Rune, who won the tournament three years ago, was treated for pain in his left hamstring, and the Dane played the rest of the match with a bandage on his leg.
Rune seemed physically handicapped at times, but was able to pull through and secure the ticket for the semi-final.
Holger Rune played well in his own service games in the first set, where he delivered one amazing serve after another.
However, the Dane was unable to break Etcheverry, so the set had to be decided by a tiebreak where the Argentinian showed the greatest composure, and he grabbed the first set against a frustrated Rune.
In the second set, Etcheverry struggled to contain Rune's strong baseline strokes, which secured the Dane a 5-3 lead.
From here, Rune broke the Argentine's serve to restore parity and force the game into a decisive third set.
The 22-year-old Dane made a perfect start to the deciding set by holding his serve and breaking Etcheverry with a string of powerful returns. With two aces, Rune also made it 3-0.
Along the way, Rune received treatment for minor thigh problems and played with a large bandage midway through the set.
The Dane seemed more inhibited, and Etcheverry managed to bring the game back to level terms at 3-3.
With great determination, Rune settled into a solid rhythm and, in the end, broke Etcheverry in the third set to win the deciding set 6-4.