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Slovakia coach Vladimir Weiss shed happy tears after the epic win.
"This is a fantastic day for Slovakia and the second greatest day of my life after the birth of my son," he said.
Paraguay topped Group F with five points following a scoreless draw with the Kiwi side who collected three points, one behind Slovakia. Italy were last with two points from as many draws.
Japan joined fellow Asians South Korea in last 16 thanks to early goals by Keisuke Honda and Endo.
Denmark, who had never failed to reach the second round in their World Cup appearances, fought end to end with Japan before Honda opened the Japanese account in the 17th minute.
A free-kick by the CSKA Moscow midfielder about 30 meters from the goal flew over the defense line of the Danish Dynamite and slipped away from the fingertips of goalkeeper Thomas Sorensen.
At half an hour, with Sorensen expecting Honda to take a set piece, Endo stunned him by curling the ball around the wall and inside the post.
Moments after the restart, Endo sent a looping but tame free-kick to goal but Sorensen fumbled it over his head and the ball smacked off the post.