Switzerland pulled off a surprise by eliminating Finland 3-2 and defending champion Sweden edged out Latvia 3-2.
Kane claimed the third-period winner to take the outright lead in the scoring table on 19 points, a U.S. record, with eight goals and 11 assists and set up a semifinal against Sweden on Saturday.
"It's my job to produce," Kane said. "It's always nice to contribute offensively."
The U.S. is looking for its first medal since picking up bronze in 2015.
"We came here to put ourselves in a position to try to win the gold," Kane said. "We're on the right path."
The U.S. took control with a couple of goals in the span of 1:43 midway through the first period in Herning.
Kane beat goaltender Pavel Francouz from the right circle before Nick Bonino fed Cam Atkinson in front of the net to stretch the lead with a backhand shot.
The Czechs hit back in the second period. Michal Repik reduced the advantage on a slap shot and Martin Necas netted the tying goal on a power play.
"It's a pity," Czech forward Tomas Plekanec said. "We created enough chances to win."
In Copenhagen, Ryan O'Reilly scored 4:57 into overtime to knock out Russia while captain Connor McDavid had three assists, including on the winning goal.
Hunting its third title in four years, Canada will face Switzerland in the semis.
"Canada are a great team, they always are," defenseman Roman Josi said after his Switzerland team beat Finland for the first time since 1972.
Defenseman Colton Parayko blasted a slap shot past goaltender Igor Shestyorkin on a power play to give Canada a 1-0 lead in the first period before Ryan Nugent-Hopkins doubled the advantage on another power play.
But Alexander Barabanov and Ilya Mikheyev scored in the second period to tie the game.
Kyle Turris made it 3-2 to Canada in the third before Sergei Andronov leveled. Pierre-Luc Dubois put Canada ahead again but Russia answered with a goal from Artyom Anisimov.
Finland looked to be heading for victory after Markus Nutivaara's first-period goal, but Switzerland rallied with goals from Enzo Corvi, Joel Vermin and Gregory Hofmann in less than four minutes midway through the second.
"We didn't start the way we wanted but we reacted in the second period and played very well from then on," Josi said.
Mikko Rantanen cut the deficit to one goal in the third period.
"This wasn't what we were looking for," Finland captain Mikael Granlund said. "They had the momentum in the second period and we were not able to turn it around."
Sweden, which won all seven of its preliminary round games, beat Latvia thanks to goals from Filip Forsberg, Viktor Arvidsson and Oliver Ekman-Larsson. Teodors Blugers and Rudolfs Balcers replied.
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