Schmeichel kept out an extra-time penalty from Luka Modric during the 1-1 draw in the last 16, and the shot-stopper denied Milan Badelj and Josip Pivaric during the shoot-out.
However, Denmark went down 3-2 on penalties after Croatia goalkeeper Danijel Subasic made three saves in the dramatic finale.
Hareide, whose team’s first-minute opener through Mathias Jorgensen was cancelled out by Mario Mandzukic, was sad for his side – and particularly his number one.
“Kasper did all he could to make sure we won, both during extra time and the penalty shoot-out but our best three players missed today,” he told a news conference.
“I’m so sorry for Kasper and the whole team. They made a wonderful effort, we played well, but that’s the way it is with penalties – the brutality of football.”
Christian Eriksen, Lasse Schone and Nicolai Jorgensen were all denied by Subasic in the shoot-out.
Hareide revealed Denmark spent little time practicing spot-kicks, although they knew who would be stepping up.
“We knew the five who would take them and they practiced a bit. But no, just very quiet after training they took a few penalties,” he said.
“Lasse Schone normally takes them for Ajax and Nicolai Jorgensen normally scores for Feyenoord.
“Norwegian research has shown that your pulse, your adrenaline is like you’re in a war zone – your stress is just as high when you take a penalty.
“We had a wonderful goalkeeper in Kasper Schmeichel but it didn’t help.”
The win sent Croatia through to a quarter-final against hosts Russia, who upset Spain, in Sochi on Saturday.