Needing to win in order to have any hope of progressing, the Albiceleste contrived to go behind to Ecuador after less than a minute - only for Lionel Messi to take the game by the scruff of the neck, scoring a hat-trick to clinch a 3-1 win.
Thanks to results elsewhere that proved enough for Argentina to qualify automatically for next summer's tournament in Russia, with Peru pushed into the play-offs and Chile missing out completely.
Elsewhere in South America, goalkeeper Alisson Becker was rested as Brazil beat Chile 3-0 to end their dominant qualifying performance.
In European action, defender Kostas Manolas will be eagerly awaiting next week's World Cup play-off draw, after Greece clinched their place on Tuesday night.
With Manolas suspended from the final game of qualification, he could only watch on as Greece ultimately secured a 4-0 victory over Gibraltar - enough to secure second in Group H and the final play-off berth.
Former Roma player Vasilis Torosidis was the unlikely hero, scoring the opener and creating another as Greece won out. Greece could potentially face Italy for a spot in Russia.
Elsewhere, Edin Dzeko played a key role as Bosnia & Herzegovina beat Estonia 2-1 to end their qualifying campaign with a win.
Dzeko's side were already out of World Cup contention - while Greece's victory meant they could not finish second in the group - but the striker came on in the 65th minute and provided the assist as Izet Hajrovic scored his second of the game to take the three points.
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