Koukal, the second Czech to shoot, deked around U.S. netminder Ryan Zapolski for the only goal of the shootout as the Americans could not get a puck past Pavel Francouz, who stopped all five shots.
"It was our dream before the tournament to go that far but I really hope it is not the end we can go farther if we keep playing like this," said Francouz.
"We are playing really good as a team, we're blocking shots the guys are helping me.
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"We didn't play any easy games and I think it is making us more and more strong.
"It was a hard game, there was a lot of emotions there nobody wanted to make the mistake.
"Both teams could win today, we were just the luckier team."
The Czechs, chasing their first medal since winning bronze at the 2006 Turin Games, await the winner of Norway and Olympic Athletes of Russia who play later on Wednesday.
With the score locked at 2-2 late in the third period, Brian O'Neill had a chance to give the U.S the lead when his wrist shot clanged off the post.
Then with 1:20 left in regulation the U.S. were handed another opportunity when Vojtech Mozik was sent off for slashing but the Americans could not convert the man advantage sending the contest to overtime.
It was a nail-biting end to a seesaw quarter-final that saw plenty of end to end action.
After killing off a powerplay the United States drew first blood in the first period when Ryan Donato rifled a shot past a badly screened Francouz.
The goal was Donato's fifth of the tournament eclipsing his father Ted's four goals from the 1992 Albertville Winter Games.
The Czechs answered late in the period when Jan Kolar's rocket from the point sailed past Zapolski.
Midway through the second the Czechs took the lead on another long range strike with Tomas Kundratek loading up from just inside the blueline but the Americans struck back two minutes later with a shorthanded goal from Jim Slater.
"Anytime you get a chance to represent your country it is a special moment," U.S. captain Brian Gionta.
"We'll be disappointed but I thought we battled hard and we left everything we had out there."
(Editing by Greg Stutchbury)
