Angel Di Maria struck late as Argentina edged Switzerland to move into the quarter-finals when the match looked certain to head to penalties.
It took until the 118th minute for the first goal when di Maria scored and broke Swiss hearts late.
Switzerland created the more genuine chances in the first half as they worked tirelessly at the back to shut down Messi.
Shaqiri turned provider for the two Swiss chances that tested Romero in goal.
The first he showed sharp footwork to round the defender, before cutting back a pass for Xhaka, who saw his shot turned away by the keeper.
The next was a one-on-one chance for Drmic after being put through by Shaqiri, but the Swiss players attempt to chip the keeper feel harmlessly into his arms.
The second half was a different story as Argentina controlled the match for extended periods as the Swiss struggled to move the ball out of their own half.
Messi went close to scoring twice, but neither side could find a winner and the match headed for extra-time.
With it looking certain to head to penalties, Switzerland lost the ball close to the centre circle. The ball found its way to Messi who skipped past his marker and teed up Di Maria to score the winner in the 118th minute. Made in Barcelona, scored in Madrid; Argentina qualified for the quarter- finals.
