Pegged as a potential favorite to win the tournament at the outset, Argentina did little to solidify those praises as their group-stage performances left plenty to be desired, despite three victories.
Tuesday's clash continued the trend of less-than-stellar performances, but in the end it was Real Madrid man Di Maria's strike in the 118th minute, off an assist from Lionel Messi, that sent Alejandro Sabella's team through, despite a stout defensive performance from the Swiss.
Switzerland had a golden chance to bury a late equalizer in stoppage time as substitute Blerim Dzemaili got his head on a cross from Bayern Munich man Xherdan Shaqiri, but the strike hit the left post, came off Dzemaili and rolled out of play.
Argentina have now won six of their seven World Cup matches that have gone to extra time, while Switzerland has not won a knockout round World Cup match since 1938.
The result sets up a quarterfinal match with the winner of Tuesday's other contest between the United States and Belgium on Saturday.
More to follow ...