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.
The Czechs await the winner of Norway and Olympic Athletes of Russia who play later on Wednesday.
With the scored 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 nail-biting end to a seesaw quarter-final that saw plenty of end to end action.
Czech Republic beat USA 3-2
Image credit: Getty Images
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 came right back two minutes later with a shorthanded goal from Jim Slater.