Rune failed to convert two match points during the third-set tiebreaker to let Etcheverry back into the match. But on the 13th point, Rune hit a dead netcord for a flukey, if fortunate, point. Etcheverry double-faulted on the next point to allow Rune to advance.
Rune's semi-final opponent will be Canadian sixth seed Felix Auger-Aliassime, who outlasted Russian qualifier Alexander Shevchenko 7-6, 3-6, 7-6.
Fourth seed Hubert Hurkacz of Poland will battle Ugo Humbert of France in the other semi-final. Hurkacz rallied past Dutch opponent Tallon Griekspoor 6-7, 6-4, 6-4, while Humbert beat Swiss wild card Dominic Stricker 6-4, 2-6, 6-2.
The top four seeds all reached the semifinal round in Vienna, with top seed Daniil Medvedev of Russia needing three sets to beat countryman and eighth seed Karen Khachanov 6-3, 3-6, 6-3 in the quarters.
Medvedev will face Greek fourth seed Stefanos Tsitsipas, who outlasted Croatia's Borna Gojo 7-6, 7-5.
Number two seed Jannik Sinner of Italy won 32 of 38 first-service points (84.2 per cent) to beat seventh-seeded American Frances Tiafoe 6-3, 6-4, and Russian third seed Andrey Rublev saved all six break points he faced to beat German fifth seed Alexander Zverev 6-1, 6-7, 6-3.
