Di María takes Argentina to last-eight in gripping extra time finish
After a tense 0-0 draw at 90 minutes and a scoreless first period of extra time, Lionel Messi broke quickly and released Di María, who swept the ball home before wheeling away in delight.
Argentina had leaned heavily on Messi throughout the group stage, the number 10 scoring four of their six goals, and the forward was at the heart of their best work in the first half, prompting and prodding in front of the massed Swiss defence.
Switzerland, who were looking to return to the quarter-finals for the first time since hosting the tournament in 1954, went closest to scoring in a tight first half when impish playmaker Xherdan Shaqiri released Josip Drmic through on goal.
The tall striker shaped to shoot but wasted the opportunity with an ill-advised chip and Argentina goalkeeper Sergio Romero gathered comfortably.
With the score tied at 0-0 at halftime, the game then opened up in the second period as Argentina began to grab the momentum, and their blue and white clad fans brought the Corinthians arena to life, chanting and bouncing in unison.
The warning signs were flashing for Switzerland when Gonzalo Higuain went close with a header before Messi drove into the box and forced Diego Benaglio into a great save.
However, despite camping out in the Swiss half for long sections of the second half Argentina could not make the breakthrough and the 90 minutes ended scoreless.
Argentina looked the stronger side in extra time when Swiss legs began to tire and, just when it looked like Swedish referee Jonas Eriksson would call for a penalty shootout, Di María curled the ball past Benaglio to seal the late triumph.