Frickson Erazo headed home the first goal after 80 minutes and Felipe Caicedo doubled their lead when he finished off a quick counter attack just a minute later.
Argentina, who were without injured Barcelona striker Lionel Messi, could not find the net in spite of taking 12 shots to Ecuador's eight and dominating possession.
"We were impatient," said defender Lucas Biglia. "And of course we missed Messi."
Ecuador, who have qualified for the finals on just three occasions, troubled Argentina out wide the entire game and the home side could not impose themselves on a team 30 places below them in the FIFA rankings.
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To compound Argentina's misery, Manchester City striker Sergio Aguero limped off midway through the first half clutching his left leg and looks likely to miss their next game, away to Paraguay next Tuesday.
Aguero had an injury concern before the game but was ruled fit to play.
Ecuador will take on Bolivia at home next Tuesday.
(Reporting by Luis Ampuero, writing by Andrew Downie; Editing by Greg Stutchbury)
