Schmeichel watched his side run riot at the Aviva Stadium in Dublin, as Denmark turned an early one-goal deficit into a resounding 5-1 win in the second leg of their World Cup qualifying play-off.
Behind Christian Eriksen's brilliant hat-trick the Danes qualified for Russia next summer, but Schmeichel felt a tinge of sadness that Ireland won't be able to join them.
"I can only have admiration for the Irish as a nation," Schmeichel told RTE Sport. "When the fourth goal went in and the Irish fans started singing Fields of Athenry, it gave me goosebumps."
"I'm obviously delighted for Denmark but I'm gutted for the Irish. They are amazing fans and great people.
"But for us it has been a long time coming and tonight we showed what a good team we are."
The Leicester City goalkeeper also hailed the performance of Eriksen, who netted his side's second, third and fourth goals to seal the fate of the Irish.
"There are no literally no words to describe how happy I am he is Danish," Schmeichel said of Eriksen.
"He is one of the best players in the world but take that aside, as a guy and a person in the dressing room, he is a guy who can take the hopes of the nation on his shoulders and do magic with it.
"He has done it again tonight and it’s not for the first time."