Wednesday's game at an overcast Corinthians Arena had been billed as a duel between Robben and Lionel Messi, but instead it was the defenders on both sides who took centre-stage.
While Messi was masterfully coralled by Dutch centre-back Ron Vlaar, Robben rarely threatened to add to his tally of three tournament goals.
Set free by Wesley Sneijder on the stroke of full-time, he was denied by a magnificent sliding block from Argentine midfielder Javier Mascherano.
Robben at least found the target in the penalty shootout, after the game finished goalless following extra time, sweeping the ball into the bottom-right corner as Romero plunged the other way.
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But Romero had already saved from Vlaar by that point and when Sneijder's subsequent spot-kick met the same fate, it was left to Maxi Rodriguez to fire Argentina into a showdown with Germany in Sunday's final.
"The third-place match? Honestly, it's only the title that interested me," he said. "Losing drives me mad. It hurts."
Beaten in three World Cup finals and two semi-finals, Holland have become accustomed to encountering heartbreak at football's showpiece event, but Wednesday's defeat was particularly hard to stomach for Robben.