Angel di Maria struck the one and only goal for Argentina in the 118th minute as Switzerland were cruelly knocked out of the World Cup in Sao Paulo this afternoon.
There were very few chances in the early stages as Switzerland upset Argentina's tempo with a number of fouls.
The Swiss had the first clear-cut opening against the run of play in the 27th minute when Xherdan Shaqiri tricked his way down the byline and squared for Granit Xhaka, whose low shot was kept out by Sergio Romero.
Ottmar Hitzfeld's side finished the first half on top and had a golden opportunity to take a shock lead when Shaqiri floated into the path of Josip Drmic.
Romero hesitated and was out of position, but Drmic let him off with a half-hearted chip that was easy for the Argentine keeper to clutch.
Argentina's first decent effort came just after the hour mark when Marcos Rojo crossed for Gonzalo Higuain, whose header was tipped over the bar by Diego Benaglio.
Five minutes later Lionel Messi had his first shot of the game, dipping a volley inches over the crossbar.
The Barcelona superstar began to have more opportunities as the 90-minute mark approached and came close to breaking the deadlock with a low shot that Benaglio just about kept out.
In extra time there were few chances for either side, with Di Maria drawing an acrobatic save out of Benaglio before he won the match with two minutes remaining.
Messi drove forward and slid the ball into his path and the Real Madrid men swept home low across goal into the far corner.
Switzerland still had a great opening to send the match into a penalty shootout, but Blerim Dzemaili headed against the post from inside the six-yard box.
Next up for Argentina in the quarter-finals will be a match against either Belgium or the USA, who face off in Salvador later this evening.