Andy Murray and Cameron Norrie both lost contests
Murray, 36, recovered from a slow start to win the first set in 76 minutes.
But the Briton lost 6-7 (5-7) 6-3 6-2 to the world number 32, who secured victory in over three hours.
Meanwhile, Cameron Norrie lost 6-2 6-4 against fifth seed Alexander Zverev in his second-round match in Vienna.
Former world number one Murray has made a habit this season of being involved in some marathon matches - and the scrap against Etcheverry was another example.
Having stopped Etcheverry going 3-0 ahead, the Scot showed all of his trademark tenacity and endurance to keep his opponent locked in long rallies during a tight opening set which could have swung either way.
Murray trailed in the tie-break before creating a set point with a superb crosscourt forehand winner and converting his chance with a deft volley.
After falling a break down early in the second set, Murray could not take three break points as Etcheverry held on in a 16-minute seventh game - which proved key in bringing the match to a decider.
Almost two and a half hours were already on the clock, and both players needed time to catch their breath before Murray succumbed to a fourth break point in another long opening game.
Murray continued to pressure the 24-year-old Argentine but became increasingly exasperated - often flinging his racquet to the court - as he could not create any more chances.
Etcheverry rattled off the final three games, and Murray's fourth loss in five matches capped a disappointing night for the British singles players.
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