With Ecuador losing 2-1 in Chile, Uruguay caught them on 25 points but their inferior goal difference consigned them to a fourth successive intercontinental play-off to try and book their ticket to the Brazil finals next year.
Uruguay went ahead three times but Maxi Rodriguez, standing in as captain for the injured Lionel Messi, twice equalised for Argentina, who had already secured their World Cup spot and first place in the South American group.
The Uruguayans finished fifth in a group handing four automatic places to Argentina, Colombia, Chile and Ecuador and will meet Jordan over two legs in November for one more berth.
Uruguay went ahead after five minutes when the Argentine defence failed to clear a free kick by Luis Suarez and the ball reached the unmarked Cristian Rodriguez who scored with a rising shot.
His Argentine namesake Maxi equalised 10 minutes later in a fast-paced Rio de la Plata clasico with thrills and spills at both ends, especially in front of Sergio Romero's Argentina goal in the second half, when Augusto Fernandez chested the ball down to Rodriguez who gave Fernando Muslera no chance.
Uruguay went ahead again when Suarez went down in the box and the referee awarded them a penalty, wrongly according to TV replays, and converted from the spot with Romero getting a touch but failing to prevent the goal.
Maxi Rodriguez levelled again four minutes before halftime with another fine goal, as he turned two defenders in a jinking run infield from the wing to shoot between defender Diego Godin's legs and past the unsighted Muslera.
Edinson Cavani put Uruguay ahead for a third time five minutes into the second half when he was fed by fellow striker Suarez in a quick two-man break and fired past the outstretched Romero.
The home side hit the woodwork twice in the space of a few seconds in the 67th minute with Cristian Stuani shaking the bar with a header and Cristian Rodriguez rapping the clearance off the far post.
Maxi Rodriguez was narrowly wide of the post with a chance at a hat-trick in the 70th.
Uruguay won play-offs to reach the 2002 and 2010 finals, where they reached the last four in South Africa but were beaten by Australia in the play-off for the 2006 finals.